Inequalities — Hard Practice Quiz

A Algebra cheat sheet for Inequalities — every key formula with its symbols defined — plus a hard-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. If $x$ and $y$ are non-negative real numbers such that $x+y=10$, what is the maximum possible value of $|xy - 20|$?

    • $5$
    • $10$
    • $20$
    • $25$

    Answer: $20$

  2. For positive real numbers $a$ and $b$, if $\frac{a+b}{2} = 10$ and $|ab - 90| > 10$, what can be inferred about $a$ and $b$?

    • $ab > 100$
    • $ab < 80$
    • $ab = 100$
    • $a=b$

    Answer: $ab < 80$

  3. If $x$ is a real number such that $|x-3| < 2$ and $|x^2 - 9| > 16$, what is the range of $x$?

    • $(1, 5)$
    • $(5, \infty)$
    • $(-\infty, -5)$
    • No solution exists.

    Answer: No solution exists.

  4. Let $x$ and $y$ be positive real numbers such that $x+y=12$. If $|xy - 30| < 6$, what is the possible range for $x$?

    • $(6 - 2\sqrt{3}, 6 + 2\sqrt{3})$
    • $(6 - 2\sqrt{3}, 6) \cup (6, 6 + 2\sqrt{3})$
    • $(0, 6 - 2\sqrt{3})$
    • $(6 + 2\sqrt{3}, 12)$

    Answer: $(6 - 2\sqrt{3}, 6) \cup (6, 6 + 2\sqrt{3})$

  5. For positive real numbers $a$ and $b$, if $ab = 100$, what is the minimum value of $|a+b - 18|$?

    • $0$
    • $2$
    • $10$
    • $18$

    Answer: $2$

  6. Find the number of integers $x$ that satisfy both $|x-5| < 3$ and $|x-2| > 1$.

    • $2$
    • $3$
    • $4$
    • $5$

    Answer: $4$

  7. For positive real numbers $p$ and $q$, if $\frac{p+q}{2} = 5$ and $|pq - 25| < 1$, what can be said about $p$ and $q$?

    • $p$ and $q$ are equal.
    • $p$ and $q$ are very close to each other.
    • $p$ and $q$ are far apart.
    • $pq < 24$

    Answer: $p$ and $q$ are very close to each other.

  8. Given positive real numbers $x$ and $y$ such that $x+y=S$. If $|xy - \frac{S^2}{4}| > \epsilon$ for some small positive $\epsilon$, what does this imply about $x$ and $y$?

    • $x=y$
    • $x \ne y$
    • $xy > \frac{S^2}{4}$
    • $xy = \frac{S^2}{4}$

    Answer: $x \ne y$

  9. Let $a, b$ be positive real numbers. If $a+b=10$ and $|a-b| < 2$, what is the range of $ab$?

    • $(24, 25)$
    • $[24, 25]$
    • $(24, 25]$
    • $[24, 25)$

    Answer: $(24, 25]$

  10. Given positive real numbers $x, y, z$. If $x+y+z=15$ and $|xy - 25| < 1$, what is the maximum possible value of $z$?

    • $15 - 2\sqrt{26}$
    • $15 - 4\sqrt{6}$
    • $15 - 5\sqrt{2}$
    • $15 - 2\sqrt{24}$

    Answer: $15 - 4\sqrt{6}$

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