Inequalities — Practice Quiz

A Algebra cheat sheet for Inequalities — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Absolute Value Inequality (Less Than)

$$|x| < a \iff -a < x < a$$

Where: \(a > 0\)

Absolute Value Inequality (Greater Than)

$$|x| > a \iff x < -a \text{ or } x > a$$

Where: \(a > 0\)

AM-GM Inequality (Arithmetic Mean - Geometric Mean)

$$\frac{a+b}{2} \ge \sqrt{ab}$$

Where: \(a, b \ge 0\)

Practice quiz

  1. Which of the following intervals is the solution to the inequality $|x| < 6$?

    • $-6 < x < 6$
    • $x < -6 \text{ or } x > 6$
    • $x > 6$
    • $x < -6$

    Answer: $-6 < x < 6$

  2. Solve the inequality $|2x - 3| \le 7$.

    • $-2 \le x \le 5$
    • $x \le -2 \text{ or } x \ge 5$
    • $-5 \le x \le 2$
    • $x \le -5 \text{ or } x \ge 2$

    Answer: $-2 \le x \le 5$

  3. The solution to the inequality $|x| > 4$ is:

    • $-4 < x < 4$
    • $x < -4 \text{ or } x > 4$
    • $x > 4$
    • $x < -4$

    Answer: $x < -4 \text{ or } x > 4$

  4. Determine the solution set for the inequality $|3x + 1| \ge 10$.

    • $- \frac{11}{3} \le x \le 3$
    • $x \le - \frac{11}{3} \text{ or } x \ge 3$
    • $x \le -3 \text{ or } x \ge \frac{11}{3}$
    • $-3 \le x \le \frac{11}{3}$

    Answer: $x \le - \frac{11}{3} \text{ or } x \ge 3$

  5. According to the AM-GM inequality, for any non-negative numbers $a$ and $b$, which of the following is true?

    • $\frac{a+b}{2} < \sqrt{ab}$
    • $\frac{a+b}{2} \ge \sqrt{ab}$
    • $\frac{a+b}{2} = \sqrt{ab}$
    • $\frac{a+b}{2} \le \sqrt{ab}$

    Answer: $\frac{a+b}{2} \ge \sqrt{ab}$

  6. If $x$ is a positive real number, what is the minimum value of $x + \frac{9}{x}$?

    • $3$
    • $6$
    • $9$
    • $12$

    Answer: $6$

  7. Which absolute value inequality represents the interval $-5 < x < 1$?

    • $|x - 2| < 3$
    • $|x + 2| < 3$
    • $|x - 3| < 2$
    • $|x + 3| < 2$

    Answer: $|x + 2| < 3$

  8. The solution set $x \le -1 \text{ or } x \ge 7$ can be expressed by which absolute value inequality?

    • $|x - 3| \ge 4$
    • $|x + 3| \ge 4$
    • $|x - 4| \ge 3$
    • $|x + 4| \ge 3$

    Answer: $|x - 3| \ge 4$

  9. If $a$ and $b$ are non-negative numbers such that $a+b=12$, what is the maximum possible value of $ab$?

    • $12$
    • $24$
    • $36$
    • $48$

    Answer: $36$

  10. For positive real numbers $x$ and $y$, if $x+y=8$, which of the following statements is true regarding their product $xy$?

    • $xy$ must be less than $16$.
    • $xy$ must be greater than $16$.
    • $xy$ can be at most $16$.
    • $xy$ can be any positive value.

    Answer: $xy$ can be at most $16$.

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