Absolute Value Inequalities - TutorOne USA

Absolute Value Inequalities

Absolute Value Inequalities Concept Explanation:

An absolute value inequality is of the form $|x| \leq a$ or $|x| \geq a$ , where $a$ is a non-negative number. The two types of absolute value inequalities have different interpretations:

$|x| \leq a$ : This means $-a \leq x \leq a$ . The solution is a range of values between $-a$ and $a$ .
$|x| \geq a$ : This means $x \leq -a$ or $x \geq a$ . The solution is outside the interval $(-a, a)$ .

Diagram:

Here’s a diagram to illustrate the solutions to the inequality $|x| \leq 4$ :

The solution is all values between $-4$ and $4$ .

Hard Questions:

Q1: Solve the inequality $|x + 2| \leq 5$ .

Step-by-Step Solution:

Rewrite the absolute value inequality as a compound inequality:

$-5 \leq x + 2 \leq 5$

Subtract 2 from all parts of the inequality:

$-7 \leq x \leq 3$

Answer:

The solution is $-7 \leq x \leq 3$ .

Q2: Solve the inequality $|3x - 4| \geq 8$ .

Step-by-Step Solution:

The absolute value inequality $|3x - 4| \geq 8$ means:

$3x - 4 \geq 8 \quad \text{or} \quad 3x - 4 \leq -8$

Solve both inequalities:

$3x \geq 12$ , so $x \geq 4$
$3x \leq -4$ , so $x \leq -\frac{4}{3}$

Answer:

The solution is $x \leq -\frac{4}{3}$ or $x \geq 4$ .

Q3: Solve $|2x + 1| < 7$ .

Step-by-Step Solution:

Rewrite the inequality as:

$-7 < 2x + 1 < 7$

Subtract 1 from all parts:

$-8 < 2x < 6$

Now divide by 2:

$-4 < x < 3$

Answer:

The solution is $-4 < x < 3$ .

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