Algebra II : Solving Absolute Value Equations

Study concepts, example questions & explanations for Algebra II

varsity tutors app store varsity tutors android store

Example Questions

Example Question #21 : Solving Absolute Value Equations

Solve for :

Possible Answers:

Correct answer:

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.

First multiply both sides by 4.

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

 

Next subtract 1 from both sides.

 

Then, divide by -3.

or

 

Now subtract 1 from both sides.

 

Finally, divide both sides by -3

Example Question #22 : Solving Absolute Value Equations

Solve for :

Possible Answers:

Correct answer:

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.

First, divide both sides by 3.

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

or

Example Question #23 : Solving Absolute Value Equations

If , find .

Possible Answers:

Correct answer:

Explanation:

If , find .

Substitute negative seven in for every x.

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

Example Question #24 : Solving Absolute Value Equations

Solve the absolute value equation.

Possible Answers:

 and 

 and 

Correct answer:

 and 

Explanation:

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

FIRST

SECOND

Example Question #25 : Solving Absolute Value Equations

Solve:

Possible Answers:

 or  

Correct answer:

 or  

Explanation:

To solve:

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:

 or

2. solve each equation for :

   or  

Example Question #26 : Solving Absolute Value Equations

Solve the following equation for b:

Possible Answers:

Correct answer:

Explanation:

Solve the following equation for b:

Let's begin by subtracting 6 from both sides:

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

 

Next, add 13 to both sides

And divide by 5:

or 

Next, add 13 to both sides

And divide by 5:

So our answers are

Example Question #27 : Solving Absolute Value Equations

Solve for x:

Possible Answers:

no solution

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.

Example Question #28 : Solving Absolute Value Equations

Solve the following equation:

Possible Answers:

Correct answer:

Explanation:

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

The first one is:

  where 

The other equations is:

 where 

Example Question #29 : Solving Absolute Value Equations

Find values of  which satisfy, 

 

Possible Answers:

 No solution

Correct answer:

 No solution

Explanation:

 

 

 

Recall the general definition of absolute value, 

 

We will attempt two cases for the absolute value, 

                                   

 

 Case 1: 

Start by isolating the abolute value term, 

Replace    with  and solve for :

 

 

 

Case 2: 

 

 

Replace   with   and solve for :

 

 

 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  in the original equation we get, 

 

This is clearly not true, since both sides are not equal, this rules out  as a solution. Similiarly, using   you can show that it also fails. Therefore, there is no solution.  

 

 

 

Example Question #30 : Solving Absolute Value Equations

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

Possible Answers:

 and 

 and 

 only

 and 

 and 

Correct answer:

 and 

Explanation:

If , then  or 

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

 or 

Learning Tools by Varsity Tutors