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Algebra 1 : How to solve absolute value equations

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Example Questions

Example Question #71 : How To Solve Absolute Value Equations

Solve: $3\left | x-2\right | = 12$

Possible Answers:

$(-\infty,-2]\cup[6,\infty]$

[-2,6]

-2,6

Correct answer:

-2,6

Explanation:

First isolate the absolute value sign. We will need to split the absolute value sign into its positive and negative components.

Divide by three on both sides.

$\frac{3\left | x-2\right | }{3}= \frac{12}{3}$

Simplify both sides.

$\left | x-2\right | =4$

Eliminate the absolute value and solve.

x-2=4

Add two on both sides.

x=6

This is one solution.

To find the other solution, break up the absolute value sign and add a negative sign in front of the left quantity. The equation becomes:

-(x-2)=4

Divide by negative one on both sides to move the negative sign to the other side.

x-2=-4

Add two on both sides.

x=-2

The solutions to this absolute value are: x=-2,6 .

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Example Question #71 : How To Solve Absolute Value Equations

Solve: $\left | x-3\right |=8$

Possible Answers:

[-5,11)

(-5,11]

[-5,11]

-5,11

(-5,11)

Correct answer:

-5,11

Explanation:

In order to solve, we will need to separate the absolute value in its positive and negative components:

x-3=8

-(x-3)=8

Solve the first equation. Add three on both sides.

x=11

This is the first solution.

Solve the second equation. Divide by negative one on both sides to move the negative sign to the right side.

$\frac{-(x-3)}{-1}=\frac{8}{-1}$

x-3=-8

Add three on both sides.

x-3+3=-8+3

Simplify both sides.

x=-5

The solutions are: x=-5,11

Be careful not to enclose these two answers using interval notation because this problem is not an inequality and does not have a range of solutions!

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Example Question #73 : How To Solve Absolute Value Equations

Which of the following is equivalent to |85 -x | -55 if is less than 85?

Possible Answers:

30 -x

140+x

140-x

30+x

None of the other responses is correct.

Correct answer:

30 -x

Explanation:

If x<85 , it follows that 85 - x > 0 . It further follows that

|85 -x| = 85 - x ,

and

|85 -x | -55 = 85 -x - 55 = 30 -x .

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Example Question #71 : How To Solve Absolute Value Equations

Which of the following is equivalent to |75-x | -20 if is greater than 75?

Possible Answers:

55+x

95-x

None of the other responses is correct.

55-x

95+x

Correct answer:

None of the other responses is correct.

Explanation:

If x> 75 , then 75 - x< 0 . It follows that

|75-x | = -(75-x) = -75+x ,

and

|75-x | -20 = -75+x - 20 = -95+x

None of the choices are equivalent to this expression.

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Example Question #75 : How To Solve Absolute Value Equations

Give the solution set of the equation

|3x - 17 | = 48

Possible Answers:

$\left \{ 10 \frac{1}{3}, 21 \frac{2}{3} \right \}$

$\left \{ -10 \frac{1}{3}, 21 \frac{2}{3} \right \}$

The equation has no solution.

$\left \{ -21 \frac{2}{3}, -10 \frac{1}{3} \right \}$

$\left \{ -21 \frac{2}{3}, 10 \frac{1}{3} \right \}$

Correct answer:

$\left \{ -10 \frac{1}{3}, 21 \frac{2}{3} \right \}$

Explanation:

|3x - 17 | = 48

It follows that either 3x-17 = -48 or 3x-17 = 48 ; we solve both separately.

3x-17 = -48

3x-17 + 17 = -48 + 17

3x = -31

$3x \div 3 = -31 \div 3$

$x = -\frac{31}{3} = -10 \frac{1}{3}$

3x-17 = 48

3x-17 + 17 = 48 + 17

3x =65

$3x \div 3 =65 \div 3$

$x = -\frac{31}{3} = 21 \frac{2}{3}$

The solution set is $\left \{ -10 \frac{1}{3}, 21 \frac{2}{3} \right \}$ .

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Example Question #71 : How To Solve Absolute Value Equations

Which of the following is equivalent to 75 - |y - 13 | if is greater than 13?

Possible Answers:

62+y

88 -y

62 -y

None of the other responses is correct.

88 + y

Correct answer:

88 -y

Explanation:

If y > 13 , then y - 13 > 0 , and

|y - 13 | = y-13 .

Therefore,

75 - |y - 13 | = 75 - (y-13) = 75 -y+13 = 88 -y

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Example Question #77 : How To Solve Absolute Value Equations

Which of the following is equivalent to 88 - |45 - x | if is less than 45?

Possible Answers:

133-x

133+x

None of the other responses is correct.

43 + x

45-x

Correct answer:

43 + x

Explanation:

If x < 45 , then 45 - x > 0 , and

|45 - x | = 45 - x

Therefore,

88 - |45 - x | = 88 - (45-x) = 88 - 45 + x = 43 + x

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Example Question #78 : How To Solve Absolute Value Equations

Which of the following is equivalent to |x+65| -3 7 if is less than 65?

Possible Answers:

-x -102

None of the other responses is correct.

-x-28

-x +102

-x+28

Correct answer:

-x -102

Explanation:

If x < 65 , it follows that

x - 65 < 65 - 65 = 0 .

Since the expression between the abosolute value bars is negative, it follows that

|x+65| = -(x+65) = -x-65

and

|x+65| -3 7 = -x-65 - 37 = -x -102

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Example Question #71 : How To Solve Absolute Value Equations

Solve for x.

|2x-7|=11

Possible Answers:

x=9 and x=-2

x=4

x=3 and x=-1

x=0 and x=5

Correct answer:

x=9 and x=-2

Explanation:

To solve for we set up the two equations.

2x-7=11 and 2x-7=-11

For the first equation

2x-7+7=11+7

2x=18

$\frac{1}{2}*2x=\frac{1}{2}*18$

x=9

And for the second equation

2x-7+7=-11+7

2x=-4

$\frac{1}{2}*2x=\frac{1}{2}*(-4)$

x=-2

And so the solutions are

x=9 and x=-2

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Example Question #71 : How To Solve Absolute Value Equations

Solve: $\left | 3x-4\right |=9$

Possible Answers:

$\frac{5}{3}, -\frac{13}{3}$

$-\frac{5}{3}, -\frac{13}{3}$

$\frac{5}{3}, \frac{13}{3}$

$\frac{13}{3}$

$-\frac{5}{3}, \frac{13}{3}$

Correct answer:

$-\frac{5}{3}, \frac{13}{3}$

Explanation:

Break up the absolute value and rewrite the equation in its positive and negative components.

3x-4=9

-(3x-4) = 9

Solve the first equation. Start by adding four on both sides.

3x-4+4=9+4

Simplify.

3x=13

Divide by three on both sides.

$x=\frac{13}{3}$

This is one of the solutions.

Solve the second equation. Start by dividing a negative one on both sides.

$\frac{-(3x-4) }{-1}= \frac{9}{-1}$

Simplify both sides.

3x-4=-9

Add four on both sides.

3x-4+4=-9+4

Simplify both sides.

3x=-5

Divide by three on both sides.

$\frac{3x}{3}=\frac{-5}{3}$

The answer for this equation is: $-\frac{5}{3}$

The answers to this absolute value equation are: $-\frac{5}{3}, \frac{13}{3}$

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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 #71 : How To Solve Absolute Value Equations

Example Question #71 : How To Solve Absolute Value Equations

Example Question #73 : How To Solve Absolute Value Equations

Example Question #71 : How To Solve Absolute Value Equations

Example Question #75 : How To Solve Absolute Value Equations

Example Question #71 : How To Solve Absolute Value Equations

Example Question #77 : How To Solve Absolute Value Equations

Example Question #78 : How To Solve Absolute Value Equations

Example Question #71 : How To Solve Absolute Value Equations

Example Question #71 : How To Solve Absolute Value Equations

All Algebra 1 Resources

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