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Algebra 1 : How to solve absolute value equations

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Example Questions

Example Question #841 : Linear Equations

Solve for :

$\left | 4x-19 \right | = 31$

Possible Answers:

x = -3 or x = 12.5

x = -12.5 or x = 3

x = -12.5 or x = 12.5

x=-12.5

x=12.5

Correct answer:

x = -3 or x = 12.5

Explanation:

$\left | 4x-19 \right | = 31$ can be rewritten as the compound statement:

4x-19= - 31 or 4x-19= 31

Solve each separately to get the solution set:

4x-19= - 31

4x-19 +19 = -31+ 19

4x = -12

$4x \div 4= -12 \div 4$

x = -3

4x-19= 31

4x- 19+19 = 31+ 19

4x = 50

$4x \div 4= 50 \div 4$

x = 12.5

So either x = -3 or x = 12.5

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Example Question #11 : How To Solve Absolute Value Equations

Solve for :

| -3x + 15 | = 42

Possible Answers:

$x \in \left \{9,19 \right \}$

$x \in \left \{-9,19 \right \}$

$x \in \left \{-19,9 \right \}$

$x \in \left \{-19,19 \right \}$

$x \in \left \{-9,9 \right \}$

Correct answer:

$x \in \left \{-9,19 \right \}$

Explanation:

Rewrite | -3x + 15 | = 42 as a compound statement:

-3x + 15= -42 or -3x + 15= 42

Solve each separately:

-3x + 15= -42

-3x + 15 -15= -42-15

-3x = -57

$-3x \div (-3) = -57 \div (-3)$

x = 19

-3x + 15= 42

-3x + 15 -15= 42-15

-3x = 27

$-3x \div (-3) = 27 \div (-3)$

x = -9

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Example Question #12 : How To Solve Absolute Value Equations

Solve for:

$\left | 3x+9\right |=12$

Possible Answers:

x=7, -7

x=-7

x=1, -7

x=1

x=1, -1

Correct answer:

x=1, -7

Explanation:

Absolute value tells you how far away a number is from 0 on the number line, so you will have to find both negative and positive values to solve this absolute value equation. To do so, you must set up two different equations.

The first one will be the positive absolute value:

3x+9=12 .

The second one will be the "negative" absolute value. You simply add a negative sign to the left side of the equation:

-(3x+9)=12

Then, you solve each equation separately, leaving you with two possible answers for the value of .

x=1 or x=-7

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Example Question #13 : How To Solve Absolute Value Equations

Solve the following for :

|x+2|=5

Possible Answers:

x=3,-7

x=3

x=7

x=3,7

x=-3,7

Correct answer:

x=3,-7

Explanation:

When we take the absolute value of anything, we will always end with a positive number. So, to clear the absolute value bars, we can split this into two seperate equations. Rather than

$\left | x+2 \right |=5$

we can set two equations of

x+2=5 or -(x+2)=5

Our first equation, x+2=5 is fairly straightforward so in this equation x=3 .

Our second equation is simple to understand once we factor the minus sign. So

-(x+2)=5

becomes

-x-2=5

So add 2 to both sides. We get

-x=7

Multiply both sides by , and we see that x=-7

So, since the absolute value sign means both our equations are true, $x=3\ or\ -7$

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Example Question #14 : How To Solve Absolute Value Equations

Solve for :

| 5x - 15 | = 65

Possible Answers:

$x \in \left \{-16,-10 \right \}$

$x \in \left \{-10,16 \right \}$

There is no solution.

$x \in \left \{-16,10 \right \}$

$x \in \left \{10,16 \right \}$

Correct answer:

$x \in \left \{-10,16 \right \}$

Explanation:

Rewrite | 5x - 15 | = 65 as a compound statement:

5x - 15 = -65 or 5x - 15 = 65

Solve each separately:

5x - 15 = -65

5x - 15 + 15 = -65+ 15

5x = -50

$5x \div 5 = -50 \div 5$

x = -10

5x - 15 = 65

5x - 15 + 15 = 65+ 15

5x = 80

$5x \div 5 = 80 \div 5$

x = 16

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Example Question #11 : How To Solve Absolute Value Equations

Solve for :

| -3x - 15 | = -42

Possible Answers:

There is no solution.

$x \in \left \{-9,19 \right \}$

x = 19

$x \in \left \{-19,19 \right \}$

$x \in \left \{-9,9 \right \}$

Correct answer:

There is no solution.

Explanation:

The absolute value of a number can never be a negative number. Therefore, no value of can make | -3x-15 | = -42 a true statement.

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Example Question #16 : How To Solve Absolute Value Equations

Solve for :

| 2x + 15 | = -77

Possible Answers:

x = 31

$x \in \left \{ 31,46\right \}$

x = 31

There is no solution.

$x \in \left \{ -31,46\right \}$

$x \in \left \{ -46, 31\right \}$

Correct answer:

There is no solution.

Explanation:

The absolute value of a number can never be a negative number. Therefore, no value of can make | -3x - 15 | = -42 a true statement.

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Example Question #12 : How To Solve Absolute Value Equations

Solve for .

|x+3|=1

Possible Answers:

x=-4

$x=0\ \text{or}\ x=-3$

$x=-2\ \text{or}\ x=2$

$x=0\ \text{or}\ x=3$

$x=-2\ \text{or}\ x=-4$

Correct answer:

$x=-2\ \text{or}\ x=-4$

Explanation:

|x+3|=1

The equation involves an absolute value. First, we need to rewrite the equation with no absolute value.

$x+3=\pm 1$

We can split this equation into two possible equations.

Equation 1: x+3 =1

Equation 2: x+3 =-1

With two equations, there are two values for . Let's start with Equation 1.

x+3 =1

Subtract from both sides.

x+3-3=1-3

x=-2

That's the first value for . To get the second value for , we need to repeat the steps, but with Equation 2.

x+3 =-1

x+3-3=-1-3

x=-4

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Example Question #12 : How To Solve Absolute Value Equations

Solve for :

$\left | 2x-7 \right | -15 = -6$

Possible Answers:

$x=-8\ \textup{or }x=1$

$x=-8\textup{ or }x=8$

x=8

$x=-1\ \textup{or }x=8$

$x=-1\textup{ or }x=1$

Correct answer:

$x=-1\ \textup{or }x=8$

Explanation:

First, isolate the absolute value expression on one side.

$\left | 2x-7 \right | -15 = -6$

$\left | 2x-7 \right | -15 + 15= -6 + 15$

$\left | 2x-7 \right | =9$

Rewrite this as the compound statement:

2x-7 =-9 or 2x-7 =9

Solve each equation separately:

2x-7 =-9

2x-7+ 7 =-9 + 7

2x = -2

$2x\div 2 = -2 \div 2$

x=-1

2x-7 =9

2x-7+ 7 =9 + 7

2x = 16

$2x\div 2 = 16 \div 2$

x=8

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Example Question #18 : How To Solve Absolute Value Equations

Solve for :

$\left | 2x +1 \right |=7$

Possible Answers:

x=3, -3

x=4, -3

x=-4, 3

$x=\pm4$

Correct answer:

x=-4, 3

Explanation:

Absolute value is a function that turns whatever is inside of it positive. This means that what's inside the function, 2x+1 , might be 7, or it could have also been -7. We have to solve for both situations.

a. 2x+1=7 subtract 1 from both sides

2x=6 divide both sides by 2

x=3

b. 2x+1=-7 subtract 1 from both sides

2x=-8 divide both sides by 2

x=-4

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Algebra 1 : How to solve absolute value equations

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #841 : Linear Equations

Example Question #11 : How To Solve Absolute Value Equations

Example Question #12 : How To Solve Absolute Value Equations

Example Question #13 : How To Solve Absolute Value Equations

Example Question #14 : How To Solve Absolute Value Equations

Example Question #11 : How To Solve Absolute Value Equations

Example Question #16 : How To Solve Absolute Value Equations

Example Question #12 : How To Solve Absolute Value Equations

Example Question #12 : How To Solve Absolute Value Equations

Example Question #18 : How To Solve Absolute Value Equations

All Algebra 1 Resources

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