Precalculus : Solve Absolute Value Inequalities

Study concepts, example questions & explanations for Precalculus

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Example Question #1 : Solve Absolute Value Inequalities

Solve for 

Possible Answers:

Correct answer:

Explanation:

In order to solve this equation, we must first isolate the absolute value. In this case, we do it by dividing both sides by  which leaves us with:

When we work with absolute value equations, we're actually solving two equations. So, our next step is to set up these two equations: 

 and 

In both cases we solve for  by adding  to both sides, leaving us with

 and 

This can be rewritten as 

Example Question #2 : Solve Absolute Value Inequalities

Solve for 

Possible Answers:

Correct answer:

Explanation:

When we work with absolute value equations, we're actually solving two equations: 

 and 

Adding  to both sides leaves us with: 

 and 

Dividing by  in order to solve for  allows us to reach our solution:

 and 

Which can be rewritten as:

Example Question #3 : Solve Absolute Value Inequalities

Solve for 

Possible Answers:

Correct answer:

Explanation:

In order to solve for  we must first isolate the absolute value. In this case, we do it by dividing both sides by 2:

As with every absolute value problem, we set up our two equations:

 and 

We isolate  by adding  to both sides:

 and 

Finally, we divide by :

 and 

Example Question #4 : Solve Absolute Value Inequalities

Solve for .

Possible Answers:

Correct answer:

Explanation:

Our first step in solving this equation is to isolate the absolute value. We do this by dividing both sides by 

.

We then set up our two equations:

 and .

Subtracting 4 from both sides leaves us with

 and .

Lastly, we multiply both sides by 2, leaving us with :

 and .

Which can be rewritten as:

 

Example Question #5 : Solve Absolute Value Inequalities

Solve for

Possible Answers:

Correct answer:

Explanation:

We first need to isolate the absolute value, which we can do in two steps:

1. Add 2 to both sides:

2. Divide both sides by 4:

Our next step is to set up our two equations:

 and 

We can now solve the equations for  by subtracting both sides by 8:

 and 

and then dividing them by 5:

 and 

Which can be rewritten as: 

Example Question #6 : Solve Absolute Value Inequalities

Solve the following absolute value inequality:

Possible Answers:

Correct answer:

Explanation:

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting seven from both sides.

Next we need to set up two inequalities since the absolute value sign will make both a negative value and a positive value positive.

From here, subtract thirteen from both sides and then divide everything by four.

Example Question #7 : Solve Absolute Value Inequalities

Solve the following absolute value inequality:

Possible Answers:

Correct answer:

Explanation:

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by dividing both sides by three. 

We now have two equations:

  and 

               

                  

So, our solution is 

Example Question #8 : Solve Absolute Value Inequalities

Solve the following inequality: 

Possible Answers:

Correct answer:

Explanation:

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting two from both sides then dividing everything by three.

Since absolute value signs make both negative and positive values positive we need to set up a double inequality.

Now to solve for  subtract four from each side.

Example Question #9 : Solve Absolute Value Inequalities

Solve for :

Possible Answers:

Correct answer:

Explanation:

If , then either or based on the meaning of the absolute value function. We have to solve for both cases.

a) subtract 5 from both sides

divide by -2, which will flip the direction of the inequality

 

Even if we didn't know the rule about flipping the inequality, this answer makes sense - for example, , and .

 

b) subtract 5 from both sides

divide by -2, once again flipping the direction of the inequality

Example Question #10 : Solve Absolute Value Inequalities

Solve the absolute value inequality.

Possible Answers:

Correct answer:

Explanation:

First, simplify so that the absolute value function is by itself on one side of the inequality.

.

Note that the symbol flipps when you divide both sides by .

Next, the two inequalities that result after removing the absolute value symbols are

 and   .

When you simplify the two inequalities, you get 

 and .

Thus, the solution is 

.

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