Inequalities with Absolute Values - Examples, Exercises and Solutions

Question Types:
Inequalities with Absolute Values: Understanding inequalityInequalities with Absolute Values: Worded problemsInequalities with Absolute Values: Applying the formulaInequalities with Absolute Values: Double absolute valueInequalities with Absolute Values: Solving the problem

Inequality

When you come across signs like < < or ,> > , and even ≤≤ or ≥≥ ,you will know it is an inequality.
Inequalities define ranges of possible values rather than single solutions, whether one value is less than, greater than, or equal to another, helping to describe situations where quantities can vary within certain limits. Therefore, The result of the inequality will be a certain range of values that you will have to find.

An important rule to keep in mind: when you double or divide both sides of the operation, the sign of the inequality is reversed!

Absolute Value Inequality

Absolute value inequality is an inequality that involves the absolute value of a variable or expression. The absolute value represents the distance of a number from zero on the number line, always as a non-negative quantity. In inequalities, this means solving for a range of values that satisfy the given condition, either greater than or less than a certain value, regardless of sign. Absolute value inequalities often split into two cases, one considering the positive side and the other the negative, providing two sets of possible solutions.

We can solve absolute value inequalities in 2 2 ways:

The geometric method
  • We will draw a number line, and mark the point from which we are interested in the distance to X X .
  • Then we find the points whose distance from the relevant point is exactly the distance mentioned in the condition.
  • Now, let's pay attention to the condition: > > or < < And we will find the results.
The algebraic method
  • Step 1: Divide into two cases:
    • Case 1: Assume the expression inside the absolute value is positive.
    • Case 2: Assume the expression inside the absolute value is negative.
  • Step 2: Return to the original inequality:
    • Case 1: Remove the absolute value and solve.
    • Case 2: Remove the negative sign, solve the inequality.
  • Step 3: Find the common domain for both cases.
  • Step 4: Use a number line to mark valid ranges.
  • Step 5: Determine all values of XXX that satisfy the inequality.
Detailed explanation

Suggested Topics to Practice in Advance

  1. Absolute Value
  2. Numerical Value
  3. Absolute Value Inequalities

Practice Inequalities with Absolute Values

Question 1

Solve the following inequality:

\( 5x+8<9 \)

Question 2

Solve the inequality:


\( 5-3x>-10 \)

Question 3

What is the solution to the following inequality?

\( 10x-4≤-3x-8 \)

Question 4

Which diagram represents the solution to the inequality below? \( 5-8x<7x+3 \)

Question 5

Solve the inequality:

\( 8x+a < 3x-4 \)

Examples with solutions for Inequalities with Absolute Values

Exercise #1

Solve the following inequality:

5x+8<9

Video Solution

Step-by-Step Solution

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

 

Answer

x<\frac{1}{5}

Exercise #2

Solve the inequality:


5-3x>-10

Video Solution

Step-by-Step Solution

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

 

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

Answer

5 > x

Exercise #3

What is the solution to the following inequality?

10x−4≤−3x−8 10x-4≤-3x-8

Video Solution

Step-by-Step Solution

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

 10x−4≤−3x−8 10x-4 ≤ -3x-8

We start by organizing the sections:

10x+3x−4≤−8 10x+3x-4 ≤ -8

13x−4≤−8 13x-4 ≤ -8

13x≤−4 13x ≤ -4

Divide by 13 to isolate the X

x≤−413 x≤-\frac{4}{13}

Let's look again at the options we were asked about:

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than−413 -\frac{4}{13} , although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to−413 -\frac{4}{13} , and only smaller than it. We know it must be large and equal, so this answer is rejected.

 

Therefore, answer B is the correct one!

Answer

Exercise #4

Which diagram represents the solution to the inequality below? 5-8x<7x+3

Video Solution

Step-by-Step Solution

First, we will move the elements:

5-8x>7x+3

5-3>7x+8x
2>13x

We divide the answer by 13, and we get:

x > \frac{2}{13}

Answer

Exercise #5

Solve the inequality:

8x+a < 3x-4

Video Solution

Step-by-Step Solution

Solving an inequality equation is just like a normal equation. We start by trying to isolate the variable (X).

It is important to note that in this equation there are two variables (X and a), so we may not reach a final result.

 8x+a<3x-4

We move the sections

8x-3x<-4-a

We reduce the terms

5x<-4-a

We divide by 5

x< -a/5 -4/5

And this is the solution!

 

Answer

x < -\frac{1}{5}a-\frac{4}{5}

Question 1

Given:

\( \left|x+2\right|<3 \)

Which of the following statements is necessarily true?

Question 2

Given:

\( \left|x+4\right|>13 \)

Which of the following statements is necessarily true?

Question 3

Given:

\( \left|x-4\right|<8 \)

Which of the following statements is necessarily true?

Question 4

Given:

\( \left|x-5\right|>-11 \)

Which of the following statements is necessarily true?

Question 5

Given:

\( \left|x-5\right|>11 \)

Which of the following statements is necessarily true?

Exercise #6

Given:

\left|x+2\right|<3

Which of the following statements is necessarily true?

Video Solution

Answer

-5 < x < 1

Exercise #7

Given:

\left|x+4\right|>13

Which of the following statements is necessarily true?

Video Solution

Answer

x>9 o x<-17

Exercise #8

Given:

\left|x-4\right|<8

Which of the following statements is necessarily true?

Video Solution

Answer

-4 < x < 12

Exercise #9

Given:

\left|x-5\right|>-11

Which of the following statements is necessarily true?

Video Solution

Answer

No solution

Exercise #10

Given:

\left|x-5\right|>11

Which of the following statements is necessarily true?

Video Solution

Answer

x>16 o x<-6

Question 1

What is the solution to the inequality shown in the diagram?

-43

Question 2

Which inequality is represented by the numerical axis below?

-7-20

Question 3

Find when the inequality is satisfied:

\( -3x+15<3x<4x+8 \)

Question 4

Given:

\( ||1-4|+3|-|a|<0 \)

Which of the following statements is necessarily true?

Question 5

Given:

\( |a|-|18-9|+|4|<0 \)

Which of the following statements is necessarily true?

Exercise #11

What is the solution to the inequality shown in the diagram?

-43

Video Solution

Answer

3≤x 3 ≤ x

Exercise #12

Which inequality is represented by the numerical axis below?

-7-20

Video Solution

Answer

\( -7

Exercise #13

Find when the inequality is satisfied:

-3x+15<3x<4x+8

Video Solution

Answer

2.5 < x

Exercise #14

Given:

||1-4|+3|-|a|<0

Which of the following statements is necessarily true?

Video Solution

Answer

a<-6

Exercise #15

Given:

|a|-|18-9|+|4|<0

Which of the following statements is necessarily true?

Video Solution

Answer

-13 < a < 13