Question

Solve each equation or inequality.∣8−5x∣≥2|8-5x| ≥ 2

Accepted Answer

Recognize that the inequality involves an absolute value expression: $$|8 - 5x| \geq 2. $$Recall that for any expression $$A$$, $$|A| \geq k$$ means $$A \geq k or$$ $$A \leq -k$$ when $$k \u003e 0. $$Set up two separate inequalities based on the definition of absolute value inequalities:

1) $$8 - 5x \geq 2

2$$) $$8 - 5x \leq -2$$ Solve the first inequality $$8 - 5x \geq 2 by $$isolating $$x$$:

Subtract 8 from both sides: $$-5x \geq 2 - 8$$

Simplify the right side: $$-5x \geq -6$$

Divide both sides by $$-5$$, remembering to reverse the inequality sign because you are dividing by a negative number: $$x \leq \frac{-6}{-5}$$ Solve the second inequality $$8 - 5x \leq -2$$ similarly:

Subtract 8 from both sides: $$-5x \leq -2 - 8$$

Simplify the right side: $$-5x \leq -10$$

Divide both sides by $$-5$$, reversing the inequality sign: $$x \geq \frac{-10}{-5}$$ Combine the two solution sets from steps 3 and 4 to express the final solution as a union of intervals where $$x$$ satisfies either inequality.

Solve each equation or inequality.
$∣ 8 - 5 x ∣ \geq 2$

Verified video answer for a similar problem:

Key Concepts

Absolute Value Inequalities

Solving Linear Inequalities

Splitting Absolute Value Expressions