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Algebra II : Solving Absolute Value Equations

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Example Questions

Example Question #21 : Absolute Value

Solve for :

$\frac{ |1-3x | }{4 } = 5$

Possible Answers:

$6.\overline{3}; - 7$

$\frac{ 19 }{ 3 }; -\frac{ 19 }{ 3 }$

8; -8

$-\frac{ 19 }{3} ; 7$

-7; -8

Correct answer:

$-\frac{ 19 }{3} ; 7$

Explanation:

In order to solve the equation, we have to isolate the variable. We do this by performing the same operation to either side of the equation.

$\frac{ |1-3x | }{4 } = 5$

First multiply both sides by 4.

| 1 - 3x | = 20

The absolute value function turns whatever is inside positive, so this has two solutions. 1 - 3 x could be positive or negative 20, so we'll solve for both:

1 - 3x = 20

Next subtract 1 from both sides.

-3x = 19

Then, divide by -3.

$x = \frac{ -19 }{3}$

1 - 3x = -20

Now subtract 1 from both sides.

-3x = -21

Finally, divide both sides by -3

x = 7

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Example Question #22 : Absolute Value

Solve for :

3|4x - 11 | = 27

Possible Answers:

$3.1\overline6; -3.1\overline6$

5; -5

$- \frac{ 1}{2} ; 5$

$3.1\overline{6}; -1.\overline{3}$

$5; \frac{ 1}{2}$

Correct answer:

$5; \frac{ 1}{2}$

Explanation:

In order to solve the equation, we have to isolate the variable. We do this by performing the same operation to either side of the equation.

3|4x - 11 | = 27

First, divide both sides by 3.

|4x - 11| = 9

The absolute value function makes whatever is inside positive, so this has two potential solutions. 4x - 11 could either be positive or negative 9, so we have to solve for both:

4 x - 11 = 9

4 x = 20

x = 5

4 x - 11 = - 9

4 x = 2

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

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Example Question #2141 : Mathematical Relationships And Basic Graphs

If g(x) = |x+3| , find g(-7) .

Possible Answers:

$-\frac{1}{4}$

$\frac{1}{4}$

Correct answer:

Explanation:

If g(x) = |x+3| , find g(-7) .

Substitute negative seven in for every x.

g(-7) = |-7+3|

g(-7) = |-4|

Recall that the absolute value of a number always becomes positive.

g(-7) = 4

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Example Question #24 : Absolute Value

Solve the absolute value equation.

4 + |2x-1| = 7

Possible Answers:

x=2 and x=-1

x=-2 and x=1

x=-1

x=2

Correct answer:

x=2 and x=-1

Explanation:

4 + |2x-1| = 7

|2x-1| = 3

Since absolute values are concerened with the distance from zero, you need two equations to show both possible solutions.

FIRST

$\\2x-1=3\\ 2x=4\\ x=2$

SECOND

$\\2x-1=-3\\ 2x=-2\\ x=-1$

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Example Question #21 : Absolute Value

Solve:

$\left | x+7\right | = 3$

Possible Answers:

x = -4

x = -3

x = 3

x = -10

x = -4 or x = -10

Correct answer:

x = -4 or x = -10

Explanation:

To solve:

$\left | x+7\right | = 3$

1. To clear the absolute value sign, we must split the equation to two possible solution. One solution for the possibility of a positive sign and another for the possibility of a negative sign:

x+7 = 3 or x+7 = -3

2. solve each equation for :

x = -4 or x = -10

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Example Question #26 : Absolute Value

Solve the following equation for b:

$\left | 5b-13\right |+6=8$

Possible Answers:

$\frac{11}{5}$

$b=3, \frac{11}{5}$

Correct answer:

$b=3, \frac{11}{5}$

Explanation:

Solve the following equation for b:

$\left | 5b-13\right |+6=8$

Let's begin by subtracting 6 from both sides:

$\left | 5b-13\right |=8-6$

$\left | 5b-13\right |=2$

Next, we can get rid of the absolute value signs and make our two equations:

5b-13=2

Next, add 13 to both sides

5b=2+13

5b=15

And divide by 5:

$b=\frac{15}{5}=3$

b=3

5b-13=-2

Next, add 13 to both sides

5b=11

And divide by 5:

$b=\frac{11}{5}$

So our answers are

$b=3, \frac{11}{5}$

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Example Question #21 : Solving Absolute Value Equations

Solve for x:

|2x-3|=-1

Possible Answers:

no solution

1,2

Correct answer:

no solution

Explanation:

Since the absolute value function only produces positive answers, an absolute value can never be equal to a negative number.

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Example Question #2152 : Mathematical Relationships And Basic Graphs

Solve the following equation:

$\small \left | x+4\right |=5$

Possible Answers:

$\small x=-1, x=9$

$\small \small x=-9, x=-1$

$\small x=-9, x=1$

$\small x=-9$

$\small x=1$

Correct answer:

$\small x=-9, x=1$

Explanation:

To solve this you need to set up two different equations then solve for x.

The first one is:

$\small x+4=5$ where $\small x=1$

The other equations is:

$\small x+4=-5$ where $\small x=-9$

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Example Question #22 : Absolute Value

Find values of which satisfy,

$2x - \left |3x+7 \right | = 4$

Possible Answers:

$x =- \frac{11}{5}, 11$

No solution

$x= {-11, -\frac{3}{5}}$

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

$x = 11, \frac{3}{5}$

Correct answer:

No solution

Explanation:

$2x - \left |3x+7 \right | = 4$

Recall the general definition of absolute value,

$\left |A\right| = \begin{cases} &+A \text{ if } A\geq 0 \\ & -A\text{ if } A<0 \end{cases}$

We will attempt two cases for the absolute value,

|3x+7| = 3x+7

|3x+7| = -(3x+7)

Case 1:

Start by isolating the abolute value term,

$\left| 3x+7 \right| = 2x -4$

Replace |3x+7| with 3x+7 and solve for :

3x+7 = 2x-4

Case 2:

$\left| 3x+7 \right| = 2x -4$

Replace |3x+7| with -(3x+7) and solve for :

-(3x+7)=2x-4

-3x - 7 = 2x -4

-5x =3

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

Testing the Solutions

Whenever solving equations with an absolute value it is crucuial to check if the solutions work in the original equation. It often occurs that one or both of the solutions will not satisfy the original equation.

For instance, if we test x = -11 in the original equation we get,

$\left| 3(-11)+7 \right| = 2(-11) -4$

26 = -26

This is clearly not true, since both sides are not equal, this rules out x=-11 as a solution. Similiarly, using $x= -\frac{3}{5}$ you can show that it also fails. Therefore, there is no solution.

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Example Question #23 : Absolute Value

What are all the possible values of that fulfill the equation below?

$\left | 3x-7\right |=15$

Possible Answers:

$\frac{8}{3}$ and $\frac{22}{3}$

$\frac{22}{3}$ only

$\frac{22}{3}$ and $-\frac{8}{3}$

$-\frac{22}{3}$ and $\frac{8}{3}$

$-\frac{8}{3}$ and $\frac{8}{3}$

Correct answer:

$\frac{22}{3}$ and $-\frac{8}{3}$

Explanation:

If $\left | 3x-7\right |=15$ , then 3x-7=15 or 3x-7=-15

Solve each of those equations to find the possible values for x.

$x=\frac{22}{3}$ or $x=-\frac{8}{3}$

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Algebra II : Solving Absolute Value Equations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Absolute Value

Example Question #22 : Absolute Value

Example Question #2141 : Mathematical Relationships And Basic Graphs

Example Question #24 : Absolute Value

Example Question #21 : Absolute Value

Example Question #26 : Absolute Value

Example Question #21 : Solving Absolute Value Equations

Example Question #2152 : Mathematical Relationships And Basic Graphs

Example Question #22 : Absolute Value

Case 1:

Case 2:

Testing the Solutions

Example Question #23 : Absolute Value

All Algebra II Resources

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