Absolute Value and Inequality - Examples, Exercises and Solutions

Understanding Absolute Value and Inequality

Complete explanation with examples

An absolute value is the distance from the zero point,
that is, it does not refer to the sum of the number (whether negative or positive),
but it focuses on how far it is from the point 0 0 .

The absolute value is symbolized as follows : ││ ││

Generally we can write:
X=X=X│-X│= │X│= X

Detailed explanation

Practice Absolute Value and Inequality

Test your knowledge with 23 quizzes

Given:

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

Which of the following statements is necessarily true?

Examples with solutions for Absolute Value and Inequality

Step-by-step solutions included
Exercise #1

Find the absolute value inequality representation for:

x+35 |x + 3| \leq 5

Step-by-Step Solution

To solve the inequality x+35 |x + 3| \leq 5 , we first consider the definition of absolute value inequality AB |A| \leq B , which is equivalent to BAB -B \leq A \leq B .

Applying this definition, we have:

5x+35 -5 \leq x + 3 \leq 5

Next, we isolate x by subtracting 3 from all parts of the inequality:

53x+3353 -5 - 3 \leq x + 3 - 3 \leq 5 - 3

This simplifies to:

8x2 -8 \leq x \leq 2

Answer:

8x2 -8 \leq x \leq 2

Exercise #2

Given:

2x+1>7 |2x + 1| > 7

Which of the following statements is necessarily true?

Step-by-Step Solution

To solve the inequality 2x+1>7 |2x + 1| > 7 , we split it into two separate inequalities:

2x+1>7 2x + 1 > 7 or 2x+1<7 2x + 1 < -7 .

For the first inequality 2x+1>7 2x + 1 > 7 , subtract 1 from both sides:

2x>6 2x > 6

Divide by 2:

x>3 x > 3

For the second inequality 2x+1<7 2x + 1 < -7 , subtract 1 from both sides:

2x<8 2x < -8

Divide by 2:

x<4 x < -4

Therefore, the solution is x>3 x > 3 or x<4 x < -4 .

Answer:

x > 3 or x < -4

Exercise #3

Given:

3x24 |3x - 2| \geq 4

Which of the following statements is necessarily true?

Step-by-Step Solution

To solve the inequality 3x24 |3x - 2| \geq 4 , we separate it into:

3x24 3x - 2 \geq 4 or 3x24 3x - 2 \leq -4 .

For 3x24 3x - 2 \geq 4 , add 2 to both sides:

3x6 3x \geq 6

Divide by 3:

x2 x \geq 2

For 3x24 3x - 2 \leq -4 , add 2 to both sides:

3x2 3x \leq -2

Divide by 3:

x23 x \leq -\frac{2}{3}

Therefore, the solution is x2 x \geq 2 or x23 x \leq -\frac{2}{3} .

Answer:

x2 x \geq 2 or x23 x \leq -\frac{2}{3}

Exercise #4

Given:

\left|2x + 1\right| > 3

Which of the following statements is necessarily true?

Step-by-Step Solution

To solve \left| 2x + 1 \right| > 3 , consider the two cases for the absolute value: 2x + 1 > 3 and 2x + 1 < -3 .

1. Solving 2x + 1 > 3 :

2x + 1 > 3

Subtract 1 from both sides:

2x > 2

Divide both sides by 2:

x > 1

2. Solving 2x + 1 < -3 :

2x + 1 < -3

Subtract 1 from both sides:

2x < -4

Divide both sides by 2:

x < -2

Thus, the solution is x < -2 \text{ or } x > 1 .

Answer:

x < -2 \text{ or } x > 1

Exercise #5

Given:

3x45 \left|3x - 4\right| \leq 5

Which of the following statements is necessarily true?

Step-by-Step Solution

To solve 3x45 \left| 3x - 4 \right| \leq 5 , we should consider two scenarios for the absolute value: 3x45 3x - 4 \leq 5 and 3x45 3x - 4 \geq -5 .

1. Solving 3x45 3x - 4 \leq 5 :

3x45 3x - 4 \leq 5

Add 4 to both sides:

3x9 3x \leq 9

Divide both sides by 3:

x3 x \leq 3

2. Solving 3x45 3x - 4 \geq -5 :

3x45 3x - 4 \geq -5

Add 4 to both sides:

3x1 3x \geq -1

Divide both sides by 3:

x13 x \geq -\frac{1}{3}

Combining both results, we have 13x3 -\frac{1}{3} \leq x \leq 3 , which is the correct answer.

Answer:

13x3 -\frac{1}{3} \leq x \leq 3

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