Module 6: Inequalities

6.3 – equations and inequalities with absolute value, learning objectives, equations with one absolute value, equations with two absolute values.

  • Recognize when a linear equation that contains absolute value does not have a solution

(6.3.2) – Solve inequalities containing absolute values

(6.3.1) – solve equations containing absolute values.

Next, we will learn how to solve an absolute value equation . To solve an equation such as [latex]|2x - 6|=8[/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is [latex]8[/latex] or [latex]-8[/latex]. This leads to two different equations we can solve independently.

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

A General Note: Absolute Value Equations

The absolute value of x is written as [latex]|x|[/latex]. It has the following properties:

For real numbers [latex]A[/latex] and [latex]B[/latex], an equation of the form [latex]|A|=B[/latex], with [latex]B\ge 0[/latex], will have solutions when [latex]A=B[/latex] or [latex]A=-B[/latex]. If [latex]B<0[/latex], the equation [latex]|A|=B[/latex] has no solution.

An absolute value equation in the form [latex]|ax+b|=c[/latex] has the following properties:

How To: Given an absolute value equation, solve it.

  • Isolate the absolute value expression on one side of the equal sign.
  • If [latex]c>0[/latex], write and solve two equations: [latex]ax+b=c[/latex] and [latex]ax+b=-c[/latex].

In the next video, we show examples of solving a simple absolute value equation.

Example: Solving Absolute Value Equations

Solve the following absolute value equations:

  • [latex]|6x+4|=8[/latex]
  • [latex]|3x+4|=-9[/latex]
  • [latex]|3x - 5|-4=6[/latex]
  • [latex]|-5x+10|=0[/latex]

Write two equations and solve each:

[latex]\begin{array}{ll}6x+4\hfill&=8\hfill& 6x+4\hfill&=-8\hfill \\ 6x\hfill&=4\hfill& 6x\hfill&=-12\hfill \\ x\hfill&=\frac{2}{3}\hfill& x\hfill&=-2\hfill \end{array}[/latex]

The two solutions are [latex]x=\frac{2}{3}[/latex], [latex]x=-2[/latex].

b. [latex]|3x+4|=-9[/latex]

There is no solution as an absolute value cannot be negative.

c. [latex]|3x - 5|-4=6[/latex]

Isolate the absolute value expression and then write two equations.

There are two solutions: [latex]x=5[/latex], [latex]x=-\frac{5}{3}[/latex].

d. [latex]|-5x+10|=0[/latex]

The equation is set equal to zero, so we have to write only one equation.

There is one solution: [latex]x=2[/latex].

In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.

Solve the absolute value equation: [latex]|1 - 4x|+8=13[/latex].

Some of our absolute value equations could be of the form [latex]|u|=|v|[/latex] where [latex]u[/latex] and [latex]v[/latex] are algebraic expressions. For example, [latex]|x-3|=|2x+1|[/latex].

How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, [latex]u[/latex], and a positive real number, [latex]a[/latex], if [latex]|u|=a[/latex], then [latex]u=a[/latex] or [latex]u=-a[/latex].

This leads us to the following property for equations with two absolute values:

Equations with Two Absolute Values

For any algebraic expressions, [latex]u[/latex] and [latex]v[/latex], if [latex]|u|=|v|[/latex], then:

[latex]u=v[/latex] or [latex]u=-v[/latex].

When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.

Solve: [latex]|5x-1|=|2x+3|[/latex].

[latex]\begin{array}{cccc} 5x-1 &=& 2x+3 &or& \,\,\,\, 5x-1 &=& -(2x+3) \\ 5x-1 &=& 2x+3 &or& \,\,\,\, 5x-1 &=& -2x-3 \\ 3x-1 &=& 3 &or& \,\,\,\, 7x-1 &=& -3 \\ 3x &=& 4 &or& \,\,\,\, 7x &=& -2 \\ x &=& \large \frac{4}{3} &or& \,\,\,\, x &=& \large -\frac{2}{7}\end{array}[/latex]

The two solutions are [latex] x=\frac{4}{3}[/latex] and [latex]x=-\frac{2}{7}[/latex]

Absolute value equations with no solutions

As we are solving absolute value equations it is important to be aware of special cases. An absolute value is defined as the distance from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.

Solve for [latex]x[/latex]. [latex]7+\left|2x-5\right|=4[/latex]

[latex]\begin{array}{r}7+\left|2x-5\right|=4\,\,\,\,\\\underline{\,-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-7\,}\\\left|2x-5\right|=-3\end{array}[/latex]

Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE.

Solve for [latex]x[/latex]. [latex]-\frac{1}{2}\left|x+3\right|=6[/latex]

[latex]\begin{array}{r}-\frac{1}{2}\left|x+3\right|=6\,\,\,\,\,\,\,\,\,\,\,\,\\\,\,\,\,\,\,\,\,\left(-2\right)-\frac{1}{2}\left|x+3\right|=\left(-2\right)6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left|x+3\right|=-12\,\,\,\,\,\end{array}[/latex]

Again, we have a result where an absolute value is negative!

There is no solution to this equation, or DNE.

In this last video, we show more examples of absolute value equations that have no solutions.

Let’s apply what you know about solving equations that contain absolute values and what you know about inequalities to solve inequalities that contain absolute values. Let’s start with a simple inequality.

[latex]\left|x\right|\leq 4[/latex]

This inequality is read, “the absolute value of [latex]x[/latex]   is less than or equal to 4.” If you are asked to solve for [latex]x[/latex], you want to find out what values of [latex]x[/latex]   are 4 units or less away from 0 on a number line. You could start by thinking about the number line and what values of [latex]x[/latex]   would satisfy this equation.

4 and [latex]−4[/latex] are both four units away from 0, so they are solutions. 3 and [latex]−3[/latex] are also solutions because each of these values is less than 4 units away from 0. So are 1 and [latex]−1[/latex], 0.5 and [latex]−0.5[/latex], and so on—there are an infinite number of values for [latex]x[/latex] that will satisfy this inequality.

The graph of this inequality will have two closed circles, at 4 and [latex]−4[/latex]. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.

Number line. Closed blue circles on negative 4 and 4. Blue line between closed blue circles.

The solution can be written this way:

Inequality: [latex]-4\leq x\leq4[/latex]

Interval: [latex]\left[-4,4\right][/latex]

The situation is a little different when the inequality sign is “greater than” or “greater than or equal to.” Consider the simple inequality [latex]\left|x\right|>3[/latex]. Again, you could think of the number line and what values of [latex]x[/latex] are greater than 3 units away from zero. This time, 3 and [latex]−3[/latex] are not included in the solution, so there are open circles on both of these values. 2 and [latex]−2[/latex] would not be solutions because they are not more than 3 units away from 0. But 5 and [latex]−5[/latex] would work, and so would all of the values extending to the left of [latex]−3[/latex] and to the right of 3. The graph would look like the one below.

Number line. Open blue circles on negative three and three. Blue arrow through all numbers less than negative 3. Blue arrow through all numbers greater than 3.

The solution to this inequality can be written this way:

Inequality : [latex]x<−3[/latex] or [latex]x>3[/latex].

Interval: [latex]\left(-\infty, -3\right)\cup\left(3,\infty\right)[/latex]

In the following video, you will see examples of how to solve and express the solution to absolute value inequalities involving both AND and OR.

Writing Solutions to Absolute Value Inequalities

For any positive value of [latex]a[/latex]   and [latex]x[/latex] ,  a single variable, or any algebraic expression:

Let’s look at a few more examples of inequalities containing absolute values.

Solve for [latex]x[/latex]. [latex]\left|x+3\right|\gt4[/latex]

Since this is a “greater than” inequality, the solution can be rewritten according to the “greater than” rule.

[latex] \displaystyle x+3<-4\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,x+3>4[/latex]

Solve each inequality.

[latex]\begin{array}{r}x+3<-4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x+3>4\\\underline{\,\,\,\,-3\,\,\,\,\,-3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,-3\,\,-3}\\x\,\,\,\,\,\,\,\,\,<-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\,\,\,\,\,\,>1\\\\x<-7\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x>1\,\,\,\,\,\,\,\end{array}[/latex]

Check the solutions in the original equation to be sure they work. Check the end point of the first related equation, [latex]−7[/latex] and the end point of the second related equation, 1.

[latex] \displaystyle \begin{array}{r}\,\,\,\left| x+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| x+3 \right|>4\\\left| -7+3 \right|=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 1+3 \right|=4\\\,\,\,\,\,\,\,\left| -4 \right|=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 4 \right|=4\\\,\,\,\,\,\,\,\,\,\,\,\,4=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4=4\end{array}[/latex]

Try [latex]−10[/latex], a value less than [latex]−7[/latex], and 5, a value greater than 1, to check the inequality.

[latex] \displaystyle \begin{array}{r}\,\,\,\,\,\left| x+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| x+3 \right|>4\\\left| -10+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 5+3 \right|>4\\\,\,\,\,\,\,\,\,\,\,\left| -7 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 8 \right|>4\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,7>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,8>4\end{array}[/latex]

Both solutions check!

Inequality: [latex] \displaystyle x<-7\,\,\,\,\,\text{or}\,\,\,\,\,x>1[/latex]

Interval: [latex]\left(-\infty, -7\right)\cup\left(1,\infty\right)[/latex]

x 1

Solve for [latex]y[/latex] .  [latex] \displaystyle 3\left|2y+6\right|-9<27[/latex]

Begin to isolate the absolute value by adding 9 to both sides of the inequality.

[latex] \displaystyle \begin{array}{r}3\left| 2y+6 \right|-9<27\\\underline{\,\,+9\,\,\,+9}\\3\left| 2y+6 \right|\,\,\,\,\,\,\,\,<36\end{array}[/latex]

Divide both sides by 3 to isolate the absolute value.

[latex]\begin{array}{r}\underline{3\left| 2y+6 \right|}\,<\underline{36}\\3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\\\,\,\,\,\,\,\,\,\,\left| 2y+6 \right|<12\end{array}[/latex]

Write the absolute value inequality using the “less than” rule. Subtract 6 from each part of the inequality.

[latex]\begin{array}{r}-12<2y+6<12\\\underline{\,\,-6\,\,\,\,\,\,\,\,\,\,\,\,\,-6\,\,\,-6}\\-18\,<\,2y\,\,\,\,\,\,\,\,\,<\,\,6\,\end{array}[/latex]

Divide by 2 to isolate the variable.

[latex]\begin{array}{r}\underline{-18}<\underline{2y}<\underline{\,6\,}\\2\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,2\,\,\\-9<\,\,y\,\,\,\,<\,3\end{array}[/latex]

Inequality: [latex] \displaystyle -9<\,\,y\,\,<3[/latex]

Interval: [latex]\left(-9,3\right)[/latex]

Open dot on negative 9 and open dot on 3, with a line through all numbers between 9 and 3.

Identify cases of inequalities containing absolute values that have no solutions

As with equations, there may be instances in which there is no solution to an inequality.

Solve for [latex]x[/latex] . [latex]\left|2x+3\right|+9\leq 7[/latex]

Isolate the absolute value by subtracting 9 from both sides of the inequality.

[latex] \displaystyle \begin{array}{r}\left| 2x+3 \right|+9\,\le \,\,\,7\,\,\\\underline{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-9\,\,\,\,\,-9}\\\,\,\,\,\,\,\,\left| 2x+3 \right|\,\,\,\le -2\,\end{array}[/latex]

The absolute value of a quantity can never be a negative number, so there is no solution to the inequality.

No solution

Absolute inequalities can be solved by rewriting them using compound inequalities. The first step to solving absolute inequalities is to isolate the absolute value. The next step is to decide whether you are working with an OR inequality or an AND inequality. If the inequality is greater than a number, we will use OR. If the inequality is less than a number, we will use AND. Remember that if we end up with an absolute value greater than or less than a negative number, there is no solution.

  • Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
  • Ex 1: Solve and Graph Basic Absolute Value inequalities.. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/0cXxATY2S-k . License : CC BY: Attribution
  • Ex 2: Solve and Graph Absolute Value inequalities . Authored by : James Sousa (Mathispower4u.com) . Located at : . License : CC BY: Attribution
  • Ex 4: Solve and Graph Absolute Value inequalities (Requires Isolating Abs. Value).. Authored by : James Sousa (Mathispower4u.com) . Located at : https://youtu.be/5jRUuiMUxWQ . License : CC BY: Attribution
  • College Algebra. . Authored by : Abramson, et al... Provided by : OpenStax. Located at : . License : CC BY: Attribution . License Terms : Download for free: http://cnx.org/contents/[email protected]:1/Preface
  • Ex 4: Solving Absolute Value Equations (Requires Isolating Abs. Value).. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : . License : CC BY: Attribution
  • Question ID 60839. Authored by : Alyson Day. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
  • Question ID 7667. Authored by : Tyler Wallace. License : CC BY: Attribution
  • Intermediate Algebra. Authored by : Lynn Marecek et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : Public Domain: No Known Copyright . License Terms : Download for free at http://cnx.org/contents/[email protected]

solving equations with two absolute value expressions

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Solving Simpler Absolute-Value Equations

Simpler Harder Special Case

When we take the absolute value of a number, we always end up with a positive number (or zero). Whether the input was positive or negative (or zero), the output is always positive (or zero). For instance, | 3 | = 3 , and | −3 | = 3 also.

This property — that both the positive and the negative become positive — makes solving absolute-value equations a little tricky. But once you learn the "trick", they're not so bad. Let's start with something simple:

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Solving Absolute Value Equations

Solve | x | = 3

I've pretty much already solved this, in my discussion above:

| −3 | = 3

So then x must be equal to 3 or equal to −3 .

But how am I supposed to solve this if I don't already know the answer? I will use the positive / negative property of the absolute value to split the equation into two cases, and I will use the fact that the "minus" sign in the negative case indicates "the opposite sign", not "a negative number".

For example, if I have x = −6 , then " − x " indicates "the opposite of x " or, in this case, −(−6) = +6 , a positive number. The "minus" sign in " − x " just indicates that I am changing the sign on x . It does not indicate a negative number. This distinction is crucial!

Whatever the value of x might be, taking the absolute value of x makes it positive. Since x might originally have been positive and might originally have been negative, I must acknowledge this fact when I remove the absolute-value bars. I do this by splitting the equation into two cases. For this exercise, these cases are as follows:

a. If the value of x was non-negative (that is, if it was positive or zero) to start with, then I can bring that value out of the absolute-value bars without changing its sign, giving me the equation x = 3 .

b. If the value of x was negative to start with, then I can bring that value out of the absolute-value bars by changing the sign on x , giving me the equation − x = 3 , which solves as x = −3 .

Then my solution is

x = ±3

We can, by the way, verify the above solution graphically. When we attempt to solve the absolute-value equation | x  | = 3 , we are, in effect, setting two line equations equal to each other and finding where they cross. For instance:

In the above, I've plotted the graph of y 1  = |  x  | (being the blue line that looks like a "V") and y 2  = 3 (being the green horizontal line). These two graphs cross at x  = −3 and at x  = +3 (being the two red dots).

If you're wanting to check your answers on a test (before you hand it in), it can be helpful to plug each side of the original absolute-value equation into your calculator as their own functions; then ask the calculator for the intersection points.

Of course, any solution can also be verified by plugging it back into the original exercise, and confirming that the left-hand side (LHS) of the equation simplifies to the same value as does the right-hand side (RHS) of the equation. For the equation above, here's my check:

x = −3

LHS: | x | = | −3 |

LHS: | x | = | +3 |

If you're ever in doubt about your solution to an equation, try graphing or else try plugging your solution back into the original question. Checking your work is always okay!

The step in the above, where the absolute-value equation was restated in two forms, one with a "plus" and one with a "minus", gives us a handy way to simplify things: When we have isolated the absolute value and go to take off the bars, we can split the equation into two cases; we will signify these cases by placing a "minus" on the opposite side of the equation (for one case) and a "plus" on the opposite side (for the other). Here's how this works:

Solve | x + 2 | = 7 , and check your solution(s).

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The absolute value is isolated on the left-hand side of the equation, so it's already set up for me to split the equation into two cases. To clear the absolute-value bars, I must split the equation into its two possible two cases, one each for if the contents of the absolute-value bars (that is, if the "argument" of the absolute value) is negative and if it's non-negative (that is, if it's positive or zero). To do this, I create two new equations, where the only difference between then is the sign on the right-hand side. First, I'll do the "minus" case:

x + 2 = −7

x = −9

Now I'll do the non-negative case, where I can just drop the bars and solve:

Now I need to check my solutions. I'll do this by plugging them back into the original equation, since the grader can't see me checking plots on my graphing calculator.

x = −9:

LHS: |(−9) + 2|

= |−7| = 7 = RHS

LHS: |(5) + 2|

= |7| = 7 = RHS

Both solutions check, so my answer is:

x = −9, 5

Solve | 2 x − 3 | − 4 = 3

First, I'll isolate the absolute-value part of the equation; that is, I'll get the absolute-value expression by itself on one side of the "equals" sign, with everything else on the other side:

| 2 x − 3 | − 4 = 3

| 2 x − 3 | = 7

Now I'll clear the absolute-value bars by splitting the equation into its two cases, one for each sign on the argument. First I'll do the negative case:

2 x − 3 = −7

2 x = −4

x = −2

And then I'll do the non-negative case:

2 x − 3 = 7

The exercise doesn't tell me to check, so I won't. (But, if I'd wanted to, I could have plugged "abs(2X−3)−4" and "3" into my calculator (as Y1 and Y2, respectively), and seen that the intersection points were at my x -values.) My answer is:

x = −2, 5

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solving equations with two absolute value expressions

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What is Absolute Value?

Practice questions, solving absolute value equations – methods & examples.

Solving equations containing an absolute value is as simple as working with regular linear equations . Before we can embark on solving absolute value equations, let’s take a review of what the word absolute value means.

In mathematics, the absolute value of a number refers to the distance of a number from zero, regardless of direction. The absolute value of a number x is generally represented as | x | = a, which implies that, x = + a and -a.

We say that the absolute value of a given number is the positive version of that number . For example, the absolute value of negative 5 is positive 5, and this can be written as: | − 5 | = 5.

Other examples of absolute values of numbers include:  |− 9| = 9, |0| = 0, − |−12| = −12 etc. From these examples of absolute values, we simply define absolute value equations as equations containing expressions with absolute value functions.

How to Solve Absolute Value Equations?

The following are the general steps for solving equations containing absolute value functions:

  • Isolate the expression containing the absolute value function.
  • Get rid of the absolute value notation by setting up the two equations so that in the first equation, the quantity inside absolute notation is positive. In the second equation, it is negative. You will remove the absolute notation and write the quantity with its suitable sign.
  • Calculate the unknown value for the positive version of the equation.
  • Solve for the negative version of the equation, in which you will first multiply the value on the other side of the equal sign by -1, and then solve.

In addition to the above steps, there are other important rules you should keep in mind when solving absolute value equations.

  • The ∣x∣is always positive: ∣x∣ → +x.
  • In | x| = a, if the  a on the right is a positive number or zero, then there is a solution.

9

Solve the equation for x: |3 + x| − 5 = 4.

  • Isolate the absolute value expression by applying the Law of equations. This means, we add 5 to both sides of the equation to obtain;

| 3 + x | − 5 + 5 = 4 + 5

| 3 + x |= 9

  • Calculate for the positive version of the equation. Solve the equation by assuming the absolute value symbols.

| 3 +  x  | = 9 → 3 +  x  = 9

Subtract 3 from both sides of the equation.

3 – 3 + x = 9 -3

  • Now calculate for the negative version of the equation by multiplying 9 by -1.

3 +  x  | = 9 → 3 +  x  = 9 × ( −1)

Also subtract 3 from both side to isolate x.

3 -3 + x = – 9 -3

Therefore 6 and -12 are the solutions.

Solve for all real values of x such that | 3x – 4 | – 2 = 3.

  • Isolate the equation with absolute function by add 2 to both sides.

= | 3x – 4 | – 2 + 2 = 3 + 2

= | 3x – 4 |= 5

Assume the absolute signs and solve for the positive version of the equation.

| 3x – 4 |= 5→3x – 4 = 5

Add 4 to both sides of the equation.

3x – 4 + 4 = 5 + 4

Divide: 3x/3 =9/3

Now solve for the negative version by multiplying 5 by -1.

3x – 4 = 5→3x – 4 = -1(5)

3x – 4 = -5

3x – 4 + 4 = – 5 + 4

Divide by 3 on both sides.

Therefore, 3 and 1/3 are the solutions.

Solve for all real values of x: Solve | 2 x  – 3 | – 4 = 3

Add 4 to both sides.

| 2 x  – 3 | -4 = 3 →| 2 x  – 3 | = 7

Assume the absolute symbols and solve for the positive version of x.

2 x  – 3 = 7

2x – 3 + 3 = 7 + 3

Now solve for the negative version of x by multiplying 7 by -1

2 x  – 3 = 7→2 x  – 3 = -1(7)

Add 3 to both sides.

2x – 3 + 3 = – 7 + 3

x = – 2

Therefore, x  = –2, 5

Solve for all real numbers of x: | x + 2 | = 7

Already the absolute value expression is isolated, therefore assume the absolute symbols and solve.

| x + 2 | = 7 → x + 2 = 7

Subtract 2 from both sides.

x + 2 – 2 = 7 -2

Multiply 7 by -1 to solve for the negative version of the equation.

x + 2 = -1(7) → x + 2 = -7

Subtract by 2 on both sides.

x + 2 – 2 = – 7 – 2

Therefore, x = -9, 5

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Absolute value

  • Absolute value I
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A positive number can be designated by a numeral preceded by a plus sign or by a numeral without a sign. Every real number is either positive or negative. The two numerals +7 and -7 have different signs but the numerical part of each is 7. This part of the numeral designates the absolute value of the number. The absolute value of a is denoted by | a | (a vertical bar on each side of the quantity).

If we designate the absolute value of an algebraic expression such as

$$\mid x+1 \mid$$

and x has a value such that x+1 is a negative number, then the absolute value of the expression will be negative of the expression . If x+1<0,

$$\mid -7 \mid =7$$

$$\mid 7 \mid =7$$

Video lesson

Simplify $$-|7|+4+|-3|$$

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Mathematics LibreTexts

2.8: Solve Absolute Value Inequalities

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  • Page ID 114116

Learning Objectives

By the end of this section, you will be able to:

  • Solve absolute value equations
  • Solve absolute value inequalities with “less than”
  • Solve absolute value inequalities with “greater than”
  • Solve applications with absolute value

Be Prepared 2.17

Before you get started, take this readiness quiz.

Evaluate: − | 7 | . − | 7 | . If you missed this problem, review Example 1.12.

Be Prepared 2.18

Fill in < , > , < , > , or = = for each of the following pairs of numbers. ⓐ | −8 | ___ − | −8 | | −8 | ___ − | −8 | ⓑ 12 ___ − | −12 | 12 ___ − | −12 | ⓒ | −6 | ___ − 6 | −6 | ___ − 6 ⓓ − ( −15 ) ___ − | −15 | − ( −15 ) ___ − | −15 | If you missed this problem, review Example 1.12.

Be Prepared 2.19

Simplify: 14 − 2 | 8 − 3 ( 4 − 1 ) | . 14 − 2 | 8 − 3 ( 4 − 1 ) | . If you missed this problem, review Example 1.13.

Solve Absolute Value Equations

As we prepare to solve absolute value equations, we review our definition of absolute value .

Absolute Value

The absolute value of a number is its distance from zero on the number line.

The absolute value of a number n is written as | n | | n | and | n | ≥ 0 | n | ≥ 0 for all numbers.

Absolute values are always greater than or equal to zero.

We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:

−5 −5 is 5 units away from 0, so | −5 | = 5 . | −5 | = 5 .

5 5 is 5 units away from 0, so | 5 | = 5 . | 5 | = 5 .

Figure 2.6 illustrates this idea.

The figure is a number line with tick marks at negative 5, 0, and 5. The distance between negative 5 and 0 is given as 5 units, so the absolute value of negative 5 is 5. The distance between 5 and 0 is 5 units, so the absolute value of 5 is 5.

For the equation | x | = 5 , | x | = 5 , we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and −5 −5 are five units from zero on the number line. They are the solutions to the equation.

If | x | = 5 then x = −5 or x = 5 If | x | = 5 then x = −5 or x = 5

The solution can be simplified to a single statement by writing x = ± 5 . x = ± 5 . This is read, “ x is equal to positive or negative 5”.

We can generalize this to the following property for absolute value equations.

Absolute Value Equations

For any algebraic expression, u , and any positive real number, a ,

if | u | = a then u = − a or u = a if | u | = a then u = − a or u = a

Remember that an absolute value cannot be a negative number.

Example 2.68

Solve: ⓐ | x | = 8 | x | = 8 ⓑ | y | = −6 | y | = −6 ⓒ | z | = 0 | z | = 0

Try It 2.135

Solve: ⓐ | x | = 2 | x | = 2 ⓑ | y | = −4 | y | = −4 ⓒ | z | = 0 | z | = 0

Try It 2.136

Solve: ⓐ | x | = 11 | x | = 11 ⓑ | y | = −5 | y | = −5 ⓒ | z | = 0 | z | = 0

To solve an absolute value equation , we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.

Example 2.69

How to solve absolute value equations.

Solve | 5 x − 4 | − 3 = 8 . | 5 x − 4 | − 3 = 8 .

Step 1 is to isolate the absolute value expression. The difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Add 3 to both sides. The result is the absolute value of the quantity 5 x minus 4 is equal to 11.

Try It 2.137

Solve: | 3 x − 5 | − 1 = 6 . | 3 x − 5 | − 1 = 6 .

Try It 2.138

Solve: | 4 x − 3 | − 5 = 2 . | 4 x − 3 | − 5 = 2 .

The steps for solving an absolute value equation are summarized here.

Solve absolute value equations.

  • Step 1. Isolate the absolute value expression.
  • Step 2. Write the equivalent equations.
  • Step 3. Solve each equation.
  • Step 4. Check each solution.

Example 2.70

Solve 2 | x − 7 | + 5 = 9 . 2 | x − 7 | + 5 = 9 .

Try It 2.139

Solve: 3 | x − 4 | − 4 = 8 . 3 | x − 4 | − 4 = 8 .

Try It 2.140

Solve: 2 | x − 5 | + 3 = 9 . 2 | x − 5 | + 3 = 9 .

Remember, an absolute value is always positive!

Example 2.71

Solve: | 2 3 x − 4 | + 11 = 3 . | 2 3 x − 4 | + 11 = 3 .

Try It 2.141

Solve: | 3 4 x − 5 | + 9 = 4 . | 3 4 x − 5 | + 9 = 4 .

Try It 2.142

Solve: | 5 6 x + 3 | + 8 = 6 . | 5 6 x + 3 | + 8 = 6 .

Some of our absolute value equations could be of the form | u | = | v | | u | = | v | where u and v are algebraic expressions. For example, | x − 3 | = | 2 x + 1 | . | x − 3 | = | 2 x + 1 | .

How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u , and a positive real number, a , if | u | = a , | u | = a , then u = − a u = − a or u = a . u = a .

This tells us that

if | u | = | v | then u = − v or u = v if | u | = | v | then u = − v or u = v

This leads us to the following property for equations with two absolute values.

Equations with Two Absolute Values

For any algebraic expressions, u and v ,

When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.

Example 2.72

Solve: | 5 x − 1 | = | 2 x + 3 | . | 5 x − 1 | = | 2 x + 3 | .

Try It 2.143

Solve: | 7 x − 3 | = | 3 x + 7 | . | 7 x − 3 | = | 3 x + 7 | .

Try It 2.144

Solve: | 6 x − 5 | = | 3 x + 4 | . | 6 x − 5 | = | 3 x + 4 | .

Solve Absolute Value Inequalities with “Less Than”

Let’s look now at what happens when we have an absolute value inequality . Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.

Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation | x | = 5 , | x | = 5 , we saw that both 5 and −5 −5 are five units from zero on the number line. They are the solutions to the equation.

| x | = 5 x = −5 or x = 5 | x | = 5 x = −5 or x = 5

What about the inequality | x | ≤ 5 ? Figure 2.7.

The figure is a number line with negative 5, 0, and 5 displayed. There is a left bracket at negative 5 and a right bracket at 5. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x which is less than or equal to 5.

In a more general way, we can see that if | u | ≤ a , Figure 2.8.

The figure is a number line with negative a 0, and a displayed. There is a left bracket at negative a and a right bracket at a. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is less than or equal to a, then negative a is less than or equal to u which is less than or equal to a.

This result is summarized here.

Absolute Value Inequalities with < < or ≤ ≤

if | u | < a , then − a < u < a if | u | ≤ a , then − a ≤ u ≤ a if | u | < a , then − a < u < a if | u | ≤ a , then − a ≤ u ≤ a

After solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.

Example 2.73

Solve | x | < 7 . | x | < 7 . Graph the solution and write the solution in interval notation.

To verify, check a value in each section of the number line showing the solution. Choose numbers such as −8 , −8 , 1, and 9.

The figure is a number line with a left parenthesis at negative 7, a right parenthesis at 7 and shading between the parentheses. The values negative 8, 1, and 9 are marked with points. The absolute value of negative 8 is less than 7 is false. It does not satisfy the absolute value of x is less than 7. The absolute value of 1 is less than 7 is true. It does satisfy the absolute value of x is less than 7. The absolute value of 9 is less than 7 is false. It does not satisfy the absolute value of x is less than 7.

Try It 2.145

Graph the solution and write the solution in interval notation: | x | < 9 . | x | < 9 .

Try It 2.146

Graph the solution and write the solution in interval notation: | x | < 1 . | x | < 1 .

Example 2.74

Solve | 5 x − 6 | ≤ 4 . | 5 x − 6 | ≤ 4 . Graph the solution and write the solution in interval notation.

Try It 2.147

Solve | 2 x − 1 | ≤ 5 . | 2 x − 1 | ≤ 5 . Graph the solution and write the solution in interval notation:

Try It 2.148

Solve | 4 x − 5 | ≤ 3 . | 4 x − 5 | ≤ 3 . Graph the solution and write the solution in interval notation:

Solve absolute value inequalities with < or ≤.

| u | < a is equivalent to − a < u < a | u | ≤ a is equivalent to − a ≤ u ≤ a | u | < a is equivalent to − a < u < a | u | ≤ a is equivalent to − a ≤ u ≤ a

  • Step 3. Solve the compound inequality.
  • Step 4. Graph the solution
  • Step 5. Write the solution using interval notation.

Solve Absolute Value Inequalities with “Greater Than”

What happens for absolute value inequalities that have “greater than”? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.

We started with the inequality | x | ≤ 5 . Figure 2.9.

The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its right and a right bracket at 5 with shading to its left. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x is less than or equal to 5.

Now we want to look at the inequality | x | ≥ 5 . | x | ≥ 5 . Where are the numbers whose distance from zero is greater than or equal to five?

Again both −5 Figure 2.10.

The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its left and a left bracket at 5 with shading to its right. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is greater than or equal to 5, then x is less than or equal to negative 5 or x is greater than or equal to 5.

In a more general way, we can see that if | u | ≥ a , Figure 2.11.

The figure is a number line with negative a, 0, and a displayed. There is a right bracket at negative a that has shading to its left and a left bracket at a with shading to its right. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is greater than or equal to a, then u is less than or equal to negative a or u is greater than or equal to a.

Absolute Value Inequalities with > or ≥

Example 2.75.

Solve | x | > 4 . | x | > 4 . Graph the solution and write the solution in interval notation.

To verify, check a value in each section of the number line showing the solution. Choose numbers such as −6 , −6 , 0, and 7.

The figure is a number line with a right parenthesis at negative 4 with shading to its left and a left parenthesis at 4 shading to its right. The values negative 6, 0, and 7 are marked with points. The absolute value of negative 6 is greater than negative 4 is true. It does not satisfy the absolute value of x is greater than 4. The absolute value of 0 is greater than 4 is false. It does not satisfy the absolute value of x is greater than 4. The absolute value of 7 is less than 4 is true. It does satisfy the absolute value of x is greater than 4.

Try It 2.149

Solve | x | > 2 . | x | > 2 . Graph the solution and write the solution in interval notation.

Try It 2.150

Solve | x | > 1 . | x | > 1 . Graph the solution and write the solution in interval notation.

Example 2.76

Solve | 2 x − 3 | ≥ 5 . | 2 x − 3 | ≥ 5 . Graph the solution and write the solution in interval notation.

Try It 2.151

Solve | 4 x − 3 | ≥ 5 . | 4 x − 3 | ≥ 5 . Graph the solution and write the solution in interval notation.

Try It 2.152

Solve | 3 x − 4 | ≥ 2 . | 3 x − 4 | ≥ 2 . Graph the solution and write the solution in interval notation.

Solve absolute value inequalities with > or ≥.

| u | > a is equivalent to u < − a or u > a | u | ≥ a is equivalent to u ≤ − a or u ≥ a | u | > a is equivalent to u < − a or u > a | u | ≥ a is equivalent to u ≤ − a or u ≥ a

Solve Applications with Absolute Value

Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain tolerance of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.

Example 2.77

The ideal diameter of a rod needed for a machine is 60 mm. The actual diameter can vary from the ideal diameter by 0.075 0.075 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Try It 2.153

The ideal diameter of a rod needed for a machine is 80 mm. The actual diameter can vary from the ideal diameter by 0.009 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Try It 2.154

The ideal diameter of a rod needed for a machine is 75 mm. The actual diameter can vary from the ideal diameter by 0.05 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Access this online resource for additional instruction and practice with solving linear absolute value equations and inequalities.

  • Solving Linear Absolute Value Equations and Inequalities

Section 2.7 Exercises

Practice makes perfect.

In the following exercises, solve.

ⓐ | x | = 6 | x | = 6 ⓑ | y | = −3 | y | = −3 ⓒ | z | = 0 | z | = 0

ⓐ | x | = 4 | x | = 4 ⓑ | y | = −5 | y | = −5 ⓒ | z | = 0 | z | = 0

ⓐ | x | = 7 | x | = 7 ⓑ | y | = −11 | y | = −11 ⓒ | z | = 0 | z | = 0

ⓐ | x | = 3 | x | = 3 ⓑ | y | = −1 | y | = −1 ⓒ | z | = 0 | z | = 0

| 2 x − 3 | − 4 = 1 | 2 x − 3 | − 4 = 1

| 4 x − 1 | − 3 = 0 | 4 x − 1 | − 3 = 0

| 3 x − 4 | + 5 = 7 | 3 x − 4 | + 5 = 7

| 4 x + 7 | + 2 = 5 | 4 x + 7 | + 2 = 5

4 | x − 1 | + 2 = 10 4 | x − 1 | + 2 = 10

3 | x − 4 | + 2 = 11 3 | x − 4 | + 2 = 11

3 | 4 x − 5 | − 4 = 11 3 | 4 x − 5 | − 4 = 11

3 | x + 2 | − 5 = 4 3 | x + 2 | − 5 = 4

−2 | x − 3 | + 8 = −4 −2 | x − 3 | + 8 = −4

−3 | x − 4 | + 4 = −5 −3 | x − 4 | + 4 = −5

| 3 4 x − 3 | + 7 = 2 | 3 4 x − 3 | + 7 = 2

| 3 5 x − 2 | + 5 = 2 | 3 5 x − 2 | + 5 = 2

| 1 2 x + 5 | + 4 = 1 | 1 2 x + 5 | + 4 = 1

| 1 4 x + 3 | + 3 = 1 | 1 4 x + 3 | + 3 = 1

| 3 x − 2 | = | 2 x − 3 | | 3 x − 2 | = | 2 x − 3 |

| 4 x + 3 | = | 2 x + 1 | | 4 x + 3 | = | 2 x + 1 |

| 6 x − 5 | = | 2 x + 3 | | 6 x − 5 | = | 2 x + 3 |

| 6 − x | = | 3 − 2 x | | 6 − x | = | 3 − 2 x |

Solve Absolute Value Inequalities with “less than”

In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.

| x | < 5 | x | < 5

| x | < 1 | x | < 1

| x | ≤ 8 | x | ≤ 8

| x | ≤ 3 | x | ≤ 3

| 3 x − 3 | ≤ 6 | 3 x − 3 | ≤ 6

| 2 x − 5 | ≤ 3 | 2 x − 5 | ≤ 3

| 2 x + 3 | + 5 < 4 | 2 x + 3 | + 5 < 4

| 3 x − 7 | + 3 < 1 | 3 x − 7 | + 3 < 1

| 4 x − 3 | < 1 | 4 x − 3 | < 1

| 6 x − 5 | < 7 | 6 x − 5 | < 7

| x − 4 | ≤ −1 | x − 4 | ≤ −1

| 5 x + 1 | ≤ −2 | 5 x + 1 | ≤ −2

Solve Absolute Value Inequalities with “greater than”

| x | > 3 | x | > 3

| x | > 6 | x | > 6

| x | ≥ 2 | x | ≥ 2

| x | ≥ 5 | x | ≥ 5

| 3 x − 8 | > − 1 | 3 x − 8 | > − 1

| x − 5 | > − 2 | x − 5 | > − 2

| 3 x − 2 | > 4 | 3 x − 2 | > 4

| 2 x − 1 | > 5 | 2 x − 1 | > 5

| x + 3 | ≥ 5 | x + 3 | ≥ 5

| x − 7 | ≥ 1 | x − 7 | ≥ 1

3 | x | + 4 ≥ 1 3 | x | + 4 ≥ 1

5 | x | + 6 ≥ 1 5 | x | + 6 ≥ 1

In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.

2 | x + 6 | + 4 = 8 2 | x + 6 | + 4 = 8

| 3 x − 4 | ≥ 2 | 3 x − 4 | ≥ 2

| 2 x − 5 | + 2 = 3 | 2 x − 5 | + 2 = 3

| 4 x − 3 | < 5 | 4 x − 3 | < 5

| 3 x + 1 | − 3 = 7 | 3 x + 1 | − 3 = 7

| 7 x + 2 | + 8 < 4 | 7 x + 2 | + 8 < 4

5 | 2 x − 1 | − 3 = 7 5 | 2 x − 1 | − 3 = 7

| 8 − x | = | 4 − 3 x | | 8 − x | = | 4 − 3 x |

| x − 7 | > − 3 | x − 7 | > − 3

A chicken farm ideally produces 200,000 eggs per day. But this total can vary by as much as 25,000 eggs. What is the maximum and minimum expected production at the farm?

An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?

In order to insure compliance with the law, Miguel routinely overshoots the weight of his tortillas by 0.5 gram. He just received a report that told him that he could be losing as much as $100,000 per year using this practice. He now plans to buy new equipment that guarantees the thickness of the tortilla within 0.005 inches. If the ideal thickness of the tortilla is 0.04 inches, what thickness of tortillas will be guaranteed?

At Lilly’s Bakery, the ideal weight of a loaf of bread is 24 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?

Writing Exercises

Write a graphical description of the absolute value of a number.

In your own words, explain how to solve the absolute value inequality, | 3 x − 2 | ≥ 4 . | 3 x − 2 | ≥ 4 .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and five rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was solve absolute value equations. In row 3, the I can was solve absolute value inequalities with “less than.” In row 4, the I can was solve absolute value inequalities with “greater than.” In row 5, the I can was solve applications with absolute value.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

IMAGES

  1. Solving Equations with Two Absolute Values

    solving equations with two absolute value expressions

  2. 24.2 Solving Absolute Value Equations Containing Two Absolute Value

    solving equations with two absolute value expressions

  3. How To Solve Absolute Value Equations

    solving equations with two absolute value expressions

  4. PPT

    solving equations with two absolute value expressions

  5. Solving Absolute Value Equations Containing TWO Absolute Value Expressions

    solving equations with two absolute value expressions

  6. Solving Absolute Value Equations (solutions, examples, videos

    solving equations with two absolute value expressions

VIDEO

  1. Solving Equations Containing Absolute Value

  2. Solving Absolute Value Equations

  3. Absolute Value Expressions in 2 minutes

  4. Solving Absolute Value Equations Part 2

  5. Solving Absolute Value Equations #dellymathsconcepts #fast #fsa

  6. Solving Absolute Value Equations

COMMENTS

  1. Solving Equations with Two Absolute Value Expressions

    From Thinkwell's College AlgebraChapter 2 Equations and Inequalities, Subchapter 2.4 Other Types of Equations

  2. Solving Equations with 2 Absolute Values

    Learn how to solve an equation algebraically involving 2 absolute values in this video math tutorial by Mario's Math Tutoring.0:12 Example 1 Solve |x-3| = |2...

  3. Solving Absolute Value Equations

    Solving absolute value equations is as easy as working with regular linear equations. The only additional key step that you need to remember is to separate the original absolute value equation into two parts: positive and negative ( ±) components. Below is the general approach on how to break them down into two equations:

  4. Worked example: absolute value equation with two solutions

    8 years ago how does the absolute value thing separate the equation into 2 equations • ( 8 votes) Upvote Flag Hamda Khan 8 years ago One of the equation's answer is a positive while the other equation's answer is a negative. Comment ( 4 votes) Upvote Downvote Flag Show more...

  5. Algebra

    There are two ways to define absolute value. There is a geometric definition and a mathematical definition. We will look at both. Geometric Definition In this definition we are going to think of |p| | p | as the distance of p p from the origin on a number line. Also, we will always use a positive value for distance.

  6. 2.6: Solving Absolute Value Equations and Inequalities

    Step 2: Set the argument of the absolute value equal to ± p. Here the argument is 5x − 1 and p = 6. 5x − 1 = − 6 or 5x − 1 = 6. Step 3: Solve each of the resulting linear equations. 5x − 1 = − 6 or 5x − 1 = 6 5x = − 5 5x = 7 x = − 1 x = 7 5. Step 4: Verify the solutions in the original equation. Check x = − 1.

  7. 1.2: Solving Absolute Value Equations

    The absolute value of a number is its distance from zero on the number line. The absolute value of a number n is written as | n | and | n | ≥ 0 for all numbers. Absolute values are always greater than or equal to zero. We learned that both a number and its opposite are the same distance from zero on the number line.

  8. Solving Absolute-Value Equations: A Special Case

    What if there are two absolute-value expressions? Can we use the same method? Yes, but only if there are exactly just the two absolute values, so that we can "isolate" each of them, one on either side of the equation. Let's consider the following equation: Solve | x + 2 | = | 3 − x |. MathHelp.com Solving Absolute Value Equations

  9. Intro to absolute value equations and graphs

    To solve absolute value equations, find x values that make the expression inside the absolute value positive or negative the constant. To graph absolute value functions, plot two lines for the positive and negative cases that meet at the expression's zero. The graph is v-shaped. Created by Sal Khan and CK-12 Foundation. Questions Tips & Thanks

  10. Solving Absolute Value Equations: Complete Guide

    STEP TWO: Solve for Positive AND Solve for Negative. For step two, you have to take the original equation |x+3| = 6 and split it up into two equations, one equal to POSITIVE 6 and the other equal to NEGATIVE 6. You also get rid of the absolute value bars. Set up two equations (positive and negative) and ditch the absolute value bars.

  11. Solving an Absolute Value Equation with Two Absolute Value Expressions

    Solving an Absolute Value Equation with Two Absolute Value Expressions (Example) James Elliott 9.52K subscribers Subscribe Subscribed 771 views 3 years ago Absolute Value...

  12. 6.3

    How To: Given an absolute value equation, solve it. Isolate the absolute value expression on one side of the equal sign. If c > 0 c > 0, write and solve two equations: ax+b = c a x + b = c and ax+b =−c a x + b = − c. In the next video, we show examples of solving a simple absolute value equation.

  13. Solving Tough Absolute Value Equations

    How to Solve Tough Absolute Value Equations. In our previous encounter of solving absolute value equations, we dealt with the easy case because the problems involved can be solved in a very straightforward manner.. In tough absolute value equations, I hope you notice that there are two absolute value expressions with different arguments on one side of the equation and a constant on the other side.

  14. Solving Harder Absolute-Value Equations

    MathHelp.com. Solving Absolute Value Equations. The absolute value is isolated on the left-hand side, so I can drop the bars and split the equation into its two cases: Case 1 ("plus"): x2 − 4 x − 5 = +7. x2 − 4 x − 12 = 0. ( x − 6) ( x + 2) = 0. x = 6, x = −2. So if the quadratic is positive, there are two solution values to the ...

  15. 6.2: Solving Absolute Value Equations

    To solve absolute value equations, first consider the following two properties of absolute value: Property 1: For b > 0, | a | = b if and only if a = b or a = − b. Property 2: For any real numbers a and b, | a | = | b | if and only if a = b or a = − b.

  16. Solve absolute value equations (practice)

    Solving absolute value equations Solve absolute value equations Google Classroom What are the solutions of the following equation? − 3 | x + 5 | + 1 = 7 | x + 5 | + 8 Choose 1 answer: x = − 57 5 or x = − 43 5 A x = − 57 5 or x = − 43 5 x = − 57 10 or x = 43 5 B x = − 57 10 or x = 43 5 x = 57 10 or x = − 43 5 C x = 57 10 or x = − 43 5

  17. Solving Simpler Absolute-Value Equations

    The step in the above, where the absolute-value equation was restated in two forms, one with a "plus" and one with a "minus", gives us a handy way to simplify things: When we have isolated the absolute value and go to take off the bars, we can split the equation into two cases; we will signify these cases by placing a "minus" on the opposite side of the equation (for one case) and a "plus" on ...

  18. Solving Absolute Value Equations

    Solution. Already the absolute value expression is isolated, therefore assume the absolute symbols and solve. | x + 2 | = 7 → x + 2 = 7. Subtract 2 from both sides. x + 2 - 2 = 7 -2. x = 5. Multiply 7 by -1 to solve for the negative version of the equation. x + 2 = -1 (7) → x + 2 = -7. Subtract by 2 on both sides.

  19. 1.2: Absolute Value Equations

    1.2: Absolute Value Equations. When solving equations with absolute value, the solution may result in more than one possible answer because, recall, absolute value is just distance from zero. Since the integer has distance units from zero, and 4 has distance 4 units from zero, then there are two integers that have distance from zero, .

  20. PDF Solving Absolute Value Equations

    1.6 Solving Absolute Value Equations 39 SELF-ASSESSMENT 1 I do not understand. 2 I can do it with help. 3 I can do it on my own. 4 I can teach someone else. EXAMPLE 2 Solving a Multi-Step Absolute Value Equation Solve ∣ 3x + 9 ∣ − 10 = −4. SOLUTION First isolate the absolute value expression on one side of the equation.

  21. Absolute Value Equation Calculator

    What do you want to calculate? Calculate it! Example: 3|2x+1|+4=25 Example (Click to try) 3|2x+1|+4=25 About absolute value equations Solve an absolute value equation using the following steps: Get the absolve value expression by itself. Set up two equations and solve them separately. Absolute Value Equation Video Lesson

  22. Absolute value (Algebra 2, Equations and inequalities)

    The absolute value of a is denoted by | a | (a vertical bar on each side of the quantity). Example If we designate the absolute value of an algebraic expression such as ∣x + 1∣ ∣ x + 1 ∣ and x has a value such that x+1 is a negative number, then the absolute value of the expression will be negative of the expression . If x+1<0, ∣−7 ∣= 7 ∣ − 7 ∣= 7

  23. 2.8: Solve Absolute Value Inequalities

    To solve an absolute value equation, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.

  24. Video: Absolute Value

    Video: Absolute Value Expression | Evaluation, Simplification & Examples Video: Solving Absolute Value Functions & Equations | Rules & Examples

  25. Worked example: absolute value equations with no solution

    And at first, this looks really daunting, but the key is to just solve for this absolute value expression and then go from there. Let me just rewrite it so that the absolute value expression really jumps out. So this is 4 times the absolute value of x plus 10 plus 4 is equal to 6 times the absolute value of x plus 10 plus 10.