Exercises 2

This page allows you to practice some exercises on Algebra. If you notice that you have difficulties, we advise you to go over the corresponding sections in the book.

These questions correspond to the following sections: - Section 2.4 - Fractions - Section 2.5 - Fractional Powers - Section 2.6 - Inequalities

Question 5

Simplify the following expressions:

(a) $\frac{3}{4} + \frac{4}{3} - 1$

(b) $\frac{3}{12} - \frac{1}{24}$

(c) $3\frac{3}{5} - 1\frac{4}{5}$

(d) $\frac{3}{5} \cdot \frac{5}{6}$

(e) $\left( \frac{3}{5} \div \frac{2}{15} \right) \cdot \frac{1}{9}$

(f) $\left( \frac{2}{3} + \frac{1}{4} \right) \div \left( \frac{3}{4} + \frac{3}{2} \right)$

(g) $\frac{x}{10} - \frac{3x}{10} + \frac{17x}{10}$

(h) $\frac{b+2}{10} - \frac{3b}{15} + \frac{b}{10}$

(i) $\frac{x + 2}{3} + \frac{1 - 3x}{4}$

(j) $\frac{3}{2b} - \frac{5}{3b}$

Show answer

(a) $\frac{9}{12} + \frac{16}{12} - 1 = \frac{25}{12} - \frac{12}{12} = \frac{13}{12}$.

(b) $\frac{6}{24} - \frac{1}{24} = \frac{5}{24}$.

(c) $\frac{18}{5} - \frac{9}{5} = \frac{9}{5}$.

(d) $\frac{15}{30} = \frac{1}{2}$.

(e) $\frac{45}{10} \cdot \frac{1}{9} = \frac{45}{90} = \frac{1}{2}$.

(f) $\frac{11}{12} \div \frac{9}{4} = \frac{44}{108} = \frac{11}{27}$.

(g) $\frac{x - 3x + 17x}{10} = \frac{15x}{10} = \frac{3x}{2}.$

(h) $\frac{b + 2 - 2b + b}{10} = \frac{2}{10} = \frac{1}{5}.$

(i) $\frac{4(x+2) + 3(1-3x)}{12} = \frac{4x - 9x + 8 + 3}{12} = \frac{1}{12}(-5x + 11).$

(j) $\frac{3 \cdot 3 - 5 \cdot 2}{6b} = -\frac{1}{6b}.$

Question 6

Cancel common factors in the following expressions: (a) $\frac{325}{625}$

(b) $\frac{8a^{2}b^{3}c}{64abc^{3}}$

(c) $\frac{2a^2 - 2b^2}{3a + 3b}$

(d) $\frac{P^3 - P Q^2}{(P+Q)^2}$

Show answer

(a) $\frac{5 \cdot 5 \cdot 13}{5 \cdot 5 \cdot 5 \cdot 5} = \frac{13}{25}.$

(b) $\frac{ab^2}{8c^2}$.

(c) $\frac{2}{3}(a-b).$

(d) $\frac{P(P+Q)(P-Q)}{(P+Q)^2} = \frac{P(P-Q)}{P+Q}$.

Question 7

Simplify the following expressions:

(a) $\frac{1}{x-2} - \frac{1}{x+2}$

(b) $\frac{6x + 25}{4x + 2} - \frac{6x^2 + x - 2}{4x^2 - 1}$

(c) $\frac{1}{8ab} - \frac{1}{8b(a + 2)}$

(d) $\frac{t}{2t + 1} - \frac{t}{2t - 1}$

(e) $\left( \frac{1}{4} - \frac{1}{5} \right)^{-2}$

(f) $n - \frac{n}{1 - \frac{1}{n}}$

(g) $\frac{1}{1 + x^{p-q}} + \frac{1}{1 + x^{q-p}}$

(h) $\frac{\frac{1}{x-1} + \frac{1}{x^2 - 1}}{x - \frac{2}{x+1}}$

Show answer

(a) $\frac{4}{x^2 - 4}$.

(b) $\frac{21}{2(2x + 1)}$.

(c) $\frac{1}{4ab(a+2)}$.

(d) $\frac{-2t}{4t^2 - 1}$.

(e) $400$.

(f) $\frac{-n}{n-1}$.

(g) $1$.

(h) $\frac{1}{(x-1)^{2}}$.

Question 8

Verify that $x^2 + 2xy − 3y^2 = (x + 3y)(x − y)$, and then simplify the expression:

$\frac{x-y}{x^2 + 2xy - 3y^2} - \frac{2}{x-y} - \frac{7}{x + 3y}$

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Note that $(x+ 3y)(x-y) = x^2 + 3xy - xy - 3y^2 = x^2 + 2xy - 3y^2.$ We can multiply the last two terms by the appropriate factor to obtain the same denominator as the first term. Some simple algebra leads to $\frac{-8x}{x^2 + 2xy - 3y^2}$.

Question 9

Compute the following numbers:

(a) $\sqrt{1600}$

(b) $\sqrt{9 + 16}$

(c) $(36)^{−1/2}$

(d) $(0.49)^{1/2}$

(e) $\sqrt{1/25}$

Show answer

(a) $40$.

(b) $5.$

(c) $\frac{1}{6}.$

(d) $0.7.$

(e) $\frac{1}{5}$.

Question 10

Let $a$ and $b$ be positive numbers. Decide whether each $``?"$ should be replaced by $=$ or $\neq$. Justify your answer.

(a) $\sqrt{25 \cdot 16} \ \ ? \ \ \sqrt{25} \cdot \sqrt{16}$

(b) $\sqrt{25 + 16} \ \ ? \ \ \sqrt{25} + \sqrt{16}$

(c) $(a + b)^{1/2} \ \ ? \ \ a^{1/2} + b^{1/2}$

(d) $(a + b)^{−1/2} \ \ ? \ \ (\sqrt{a + b})^{−1}$

Show answer

(a) $=$, as both expressions equal $20$.

(b) $\neq$, as $\sqrt{25 + 16} = \sqrt{41} \neq 9 = \sqrt{25} + \sqrt{16}$.

(c) $\neq$, as this is essentially the same problem as in (b). Take $a=b=1$ as an easy counterexample.

(d) $=$, note that $(a+b)^{-1/2} = [(a+b)^{1/2}]^{-1} = (\sqrt{a+b})^{-1}$.

Question 11

Solve for $x$ the following equalities:

(a) $\sqrt{x} = 9$

(b) $\sqrt{x} \cdot \sqrt{4} = 4$

(c) $\sqrt{x + 2} = 25$

(d) $\sqrt{3} \cdot \sqrt{5} = \sqrt{x}$

(e) $2^{2−x} = 8$

(f) $2^x − 2^{x−1} = 4$

Show answer

(a) $81$.

(b) $4$.

(c) $623$.

(d) $15$.

(e) $-1$.

(f) $2^{x} - 2^{x-1} = 2^{x-1}(2 - 1) = 4$ for $x=3$.

Question 12

Rationalize the denominator and simplify the following expressions:

(a) $\frac{6}{\sqrt{7}}$

(b) $\frac{\sqrt{32}}{\sqrt{2}}$

(c) $\frac{\sqrt{54} − \sqrt{24}}{\sqrt{6}}$

(d) $\frac{2}{\sqrt{3}\sqrt{8}}$

(e) $\frac{1}{\sqrt{7} + \sqrt{5}}$

(f) $\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

(g) $\frac{x\sqrt{y} - y\sqrt{x}}{x\sqrt{y} + y\sqrt{x}}$

(h) $\frac{1 - \sqrt{x + 1}}{1 + \sqrt{x + 1}}$

Show answer

(a) $\frac{6}{7}\sqrt{7}$.

(b) $4$.

(c) $1$.

(d) $\frac{1}{6}\sqrt{6}$.

(e) $\frac{1}{2}(\sqrt{7} - \sqrt{5})$.

(f) $4 - \sqrt{15}$.

(g) $\frac{(\sqrt{x} - \sqrt{y})^{2}}{x-y}$.

(h) $\frac{1}{x}(2\sqrt{x+1} - x - 2)$.

Question 13

Simplify the following expressions, so that each contains only a single exponent:

(a) $\{ \left[ (a^{1/2})^{2/3} \right]^{3/4} \}^{4/5}$

(b) $a^{1/2} \cdot a^{2/3} \cdot a^{3/4} \cdot a^{4/5}$

(c) $\{[(3a)^{−1}]^{−2}(2a^{−2}) −1 \}/a^{−3}$

(d) $\frac{\sqrt[3]{a} \cdot a^{1/12} \cdot \sqrt[4]{a^3}}{a^{5/12} \cdot \sqrt{a}}$

Show answer

(a) $a^{1/5}$.

(b) $a^{163/60}$.

(c) $9a^{7}/2$.

(d) $a^{1/4}$.

Question 14

Decide which of the following inequalities are true:

(a) $−6.15 > −7.16$

(b) $6 ≥ 6$

(c) $(−5)^2 ≤ 0$

(d) $− \frac{1}{2} π < −\frac{1}{3}π$

(e) $\frac{4}{5} > \frac{6}{7}$

(f) $2^3 < 3^2$

(g) $2^{−3} < 3^{−2}$

(h) $\frac{1}{2} - \frac{2}{3} < \frac{1}{4} - \frac{1}{3}$

Show answer

True: (a), (b), (d), (f).

False: (c), (e), (g).

Question 15

Find what values of $x$ satisfy the following inequalities:

(a) $3x + 5 < x − 13$

(b) $3x − (x − 1) ≥ x − (1 − x)$

(c) $\frac{x}{24} - (x+1) + \frac{3x}{8} < \frac{5}{12}(x+1)$

(d) $1 ≤ \frac{1}{3}(2x − 1) + \frac{8}{3} (1 − x) < 16$

(e) $-5 < \frac{1}{x} < 0$

Show answer

(a) $x < -9$.

(b) All $x$.

(c) $x > -17/12$.

(d) $-41/6 < x \leq 2/3$.

(e) $x < -1/5$.

Question 16

Fill in each “$?$” with “$⇒$”, “$⇐$”, or “$⇔$” in order to complete a true statement:

(a) $x(x + 3) < 0 \quad ? \quad x > −3$

(b) $x^2 < 9 \quad ? \quad x < 3$

(c) $x^2 > 0 \quad ? \quad x > 0$

(d) $x > y^2 \quad ? \quad x > 0$

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(a) $\implies$

(b) $\implies$

(c) $\impliedby$

(d) $\implies$

Question 17

Decide whether the following inequalities are valid for all $x$ and $y$. If not, specify exactly the values of $x$ and $y$ for which the relation holds.

(a) $x + 1 > x$

(b) $x^2 > x$

(c) $x + x > x$

(d) $x^2 + y^2 ≥ 2xy$

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(a) Yes.

(b) No, this is not true for $x \in (-1,1)$.

(c) No, not true for $x \leq 0$.

(d) Yes, because the inequality is equivalent to $(x-y)^2 = x^2 - 2xy + y^2 \geq 0$, which holds for all $x$ and $y$.