Review Exercises and Sample Exam

1.9 Review Exercises and Sample Exam

Review Exercises

Review of Real Numbers and Absolute Value

Reduce to lowest terms.

$\frac{56}{120}$
$\frac{54}{60}$
$\frac{155}{90}$
$\frac{315}{120}$

Simplify.

$- (- \frac{1}{2})$
$- (- (- \frac{5}{8}))$
$- (- (- a))$
$- (- (- (- a)))$

Graph the solution set and give the interval notation equivalent.

$x \geq - 10$
$x < 0$
$- 8 \leq x < 0$
$- 10 < x \leq 4$
$x < 3 and x \geq - 1$
$x < 0 and x > 1$
$x < - 2 or x > - 6$
$x \leq - 1 or x > 3$

Determine the inequality that corresponds to the set expressed using interval notation.

$[- 8, \infty)$
$(- \infty, - 7)$
$[12, 32]$
$[- 10, 0)$
$(- \infty, 1] \cup (5, \infty)$
$(- \infty, - 10) \cup (- 5, \infty)$
$(- 4, \infty)$
$(- \infty, 0)$

Simplify.

$− | - \frac{3}{4} |$
$− | - (- \frac{2}{3}) |$
$- (- | - 4 |)$
$- (- (- | - 3 |))$

Determine the values represented by a.

$| a | = 6$
$| a | = 1$
$| a | = - 5$
$| a | = a$

Operations with Real Numbers

Perform the operations.

$\frac{1}{4} - \frac{1}{5} + \frac{3}{20}$
$\frac{2}{3} - (- \frac{3}{4}) - \frac{5}{12}$
$\frac{5}{3} (- \frac{6}{7}) \div (\frac{5}{14})$
$(- \frac{8}{9}) \div \frac{16}{27} (\frac{2}{15})$
${(- \frac{2}{3})}^{3}$
${(- \frac{3}{4})}^{2}$
${(- 7)}^{2} - 8^{2}$
$- 4^{2} + {(- 4)}^{3}$
$10 - 8 ({(3 - 5)}^{2} - 2)$
$4 + 5 (3 - {(2 - 3)}^{2})$
$- 3^{2} - (7 - {(- 4 + 2)}^{3})$
${(- 4 + 1)}^{2} - {(3 - 6)}^{3}$
$\frac{10 - 3 {(- 2)}^{3}}{3^{2} - {(- 4)}^{2}}$
$\frac{6 [{(- 5)}^{2} - {(- 3)}^{2}]}{4 - 6 {(- 2)}^{2}}$
$7 - 3 | 6 - {(- 3 - 2)}^{2} |$
$- 6^{2} + 5 | 3 - 2 {(- 2)}^{2} |$
$\frac{12 - | 6 - 2 {(- 4)}^{2} |}{3 - | - 4 |}$
$\frac{- {(5 - 2 | - 3 |)}^{3}}{| 4 - {(- 3)}^{2} | - 3^{2}}$

Square and Cube Roots of Real Numbers

Simplify.

$3 \sqrt{8}$
$5 \sqrt{18}$
$6 \sqrt{0}$
$\sqrt{- 6}$
$\sqrt{\frac{75}{16}}$
$\sqrt{\frac{80}{49}}$
$\sqrt[3]{40}$
$\sqrt[3]{81}$
$\sqrt[3]{- 81}$
$\sqrt[3]{- 32}$
$\sqrt[3]{\frac{250}{27}}$
$\sqrt[3]{\frac{1}{125}}$

Use a calculator to approximate the following to the nearest thousandth.

$\sqrt{12}$
$3 \sqrt{14}$
$\sqrt[3]{18}$
$7 \sqrt[3]{25}$
Find the length of the diagonal of a square with sides measuring 8 centimeters.
Find the length of the diagonal of a rectangle with sides measuring 6 centimeters and 12 centimeters.

Algebraic Expressions and Formulas

Multiply.

$\frac{2}{3} (9 x^{2} + 3 x - 6)$
$- 5 (\frac{1}{5} y^{2} - \frac{3}{5} y + \frac{1}{2})$
$(a^{2} - 5 a b - 2 b^{2}) (- 3)$
$(2 m^{2} - 3 m n + n^{2}) \cdot 6$

Combine like terms.

$5 x^{2} y - 3 x y^{2} - 4 x^{2} y - 7 x y^{2}$
$9 x^{2} y^{2} + 8 x y + 3 - 5 x^{2} y^{2} - 8 x y - 2$
$a^{2} b^{2} - 7 a b + 6 - a^{2} b^{2} + 12 a b - 5$
$5 m^{2} n - 3 m n + 2 m n^{2} - 2 n m - 4 m^{2} n + m n^{2}$

Simplify.

$5 x^{2} + 4 x - 3 (2 x^{2} - 4 x - 1)$
$(6 x^{2} y^{2} + 3 x y - 1) - (7 x^{2} y^{2} - 3 x y + 2)$
$a^{2} - b^{2} - (2 a^{2} + a b - 3 b^{2})$
$m^{2} + m n - 6 (m^{2} - 3 n^{2})$

Evaluate.

$x^{2} - 3 x + 1$ where $x = - \frac{1}{2}$
$x^{2} - x - 1$ where $x = - \frac{2}{3}$
$a^{4} - b^{4}$ where $a = - 3$ and $b = - 1$
$a^{2} - 3 a b + 5 b^{2}$ where $a = 4$ and $b = - 2$
$(2 x + 1) (x - 3)$ where $x = - 3$
$(3 x + 1) (x + 5)$ where $x = - 5$
$\sqrt{b^{2} - 4 a c}$ where $a = 2$ , $b = - 4$ , and $c = - 1$
$\sqrt{b^{2} - 4 a c}$ where $a = 3$ , $b = - 6$ , and $c = - 2$
$π r^{2} h$ where $r = 2 \sqrt{3}$ and $h = 5$
$\frac{4}{3} π r^{3}$ where $r = 2 \sqrt[3]{6}$
What is the simple interest earned on a 4 year investment of $4,500 at an annual interest rate of $4 \frac{3}{4}$ %?
James traveled at an average speed of 48 miles per hour for $2 \frac{1}{4}$ hours. How far did he travel?
The period of a pendulum T in seconds is given by the formula $T = 2 π \sqrt{\frac{L}{32}}$ where L represents its length in feet. Approximate the period of a pendulum with length 2 feet. Round off to the nearest tenth of a foot.
The average distance d, in miles, a person can see an object is given by the formula $d = \frac{\sqrt{6 h}}{2}$ where h represents the person’s height above the ground, measured in feet. What average distance can a person see an object from a height of 10 feet? Round off to the nearest tenth of a mile.

Rules of Exponents and Scientific Notation

Multiply.

$\frac{x^{10} \cdot x^{2}}{x^{5}}$
$\frac{x^{6} {(x^{2})}^{4}}{x^{3}}$
$- 7 x^{2} y z^{3} \cdot 3 x^{4} y^{2} z$
$3 a^{2} b^{3} c {(- 4 a^{2} b c^{4})}^{2}$
$\frac{- 10 a^{5} b^{0} c^{- 4}}{25 a^{- 2} b^{2} c^{- 3}}$
$\frac{- 12 x^{- 6} y^{- 2} z}{36 x^{- 3} y^{4} z^{6}}$
${(- 2 x^{- 5} y^{- 3} z)}^{- 4}$
${(3 x^{6} y^{- 3} z^{0})}^{- 3}$
${(\frac{- 5 a^{2} b^{3}}{c^{5}})}^{2}$
${(\frac{- 3 m^{5}}{5 n^{2}})}^{3}$
${(\frac{- 2 a^{- 2} b^{3} c}{3 a b^{- 2} c^{0}})}^{- 3}$
${(\frac{6 a^{3} b^{- 3} c}{2 a^{7} b^{0} c^{- 4}})}^{- 2}$

Perform the operations.

$(4.3 \times 10^{22}) (3.1 \times 10^{- 8})$
$(6.8 \times 10^{- 33}) (1.6 \times 10^{7})$
$\frac{1.4 \times 10^{- 32}}{2 \times 10^{- 10}}$
$\frac{1.15 \times 10^{26}}{2.3 \times 10^{- 7}}$
The value of a new tablet computer in dollars can be estimated using the formula $v = 450 {(t + 1)}^{- 1}$ where t represents the number of years after it is purchased. Use the formula to estimate the value of the tablet computer $2 \frac{1}{2}$ years after it was purchased.
The speed of light is approximately $6.7 \times 10^{8}$ miles per hour. Express this speed in miles per minute and determine the distance light travels in 4 minutes.

Polynomials and Their Operations

Simplify.

$(x^{2} + 3 x - 5) - (2 x^{2} + 5 x - 7)$
$(6 x^{2} - 3 x + 5) + (9 x^{2} + 3 x - 4)$
$(a^{2} b^{2} - a b + 6) - (a b + 9) + (a^{2} b^{2} - 10)$
$(x^{2} - 2 y^{2}) - (x^{2} + 3 x y - y^{2}) - (3 x y + y^{2})$
$- \frac{3}{4} (16 x^{2} + 8 x - 4)$
$6 (\frac{4}{3} x^{2} - \frac{3}{2} x + \frac{5}{6})$
$(2 x + 5) (x - 4)$
$(3 x - 2) (x^{2} - 5 x + 2)$
$(x^{2} - 2 x + 5) (2 x^{2} - x + 4)$
$(a^{2} + b^{2}) (a^{2} - b^{2})$
$(2 a + b) (4 a^{2} - 2 a b + b^{2})$
${(2 x - 3)}^{2}$
${(3 x - 1)}^{3}$
${(2 x + 3)}^{4}$
${(x^{2} - y^{2})}^{2}$
${(x^{2} y^{2} + 1)}^{2}$
$\frac{27 a^{2} b - 9 a b + 81 a b^{2}}{3 a b}$
$\frac{125 x^{3} y^{3} - 25 x^{2} y^{2} + 5 x y^{2}}{5 x y^{2}}$
$\frac{2 x^{3} - 7 x^{2} + 7 x - 2}{2 x - 1}$
$\frac{12 x^{3} + 5 x^{2} - 7 x - 3}{4 x + 3}$
$\frac{5 x^{3} - 21 x^{2} + 6 x - 3}{x - 4}$
$\frac{x^{4} + x^{3} - 3 x^{2} + 10 x - 1}{x + 3}$
$\frac{a^{4} - a^{3} + 4 a^{2} - 2 a + 4}{a^{2} + 2}$
$\frac{8 a^{4} - 10}{a^{2} - 2}$

Solving Linear Equations

Solve.

$6 x - 8 = 2$
$12 x - 5 = 3$
$\frac{5}{4} x - 3 = \frac{1}{2}$
$\frac{5}{6} x - \frac{1}{4} = \frac{3}{2}$
$\frac{9 x + 2}{3} = \frac{5}{6}$
$\frac{3 x - 8}{10} = \frac{5}{2}$
$3 a - 5 - 2 a = 4 a - 6$
$8 - 5 y + 2 = 4 - 7 y$
$5 x - 6 - 8 x = 1 - 3 x$
$17 - 6 x - 10 = 5 x + 7 - 11 x$
$5 (3 x + 3) - (10 x - 4) = 4$
$6 - 2 (3 x - 1) = - 4 (1 - 3 x)$
$9 - 3 (2 x + 3) + 6 x = 0$
$- 5 (x + 2) - (4 - 5 x) = 1$
$\frac{5}{9} (6 y + 27) = 2 - \frac{1}{3} (2 y + 3)$
$4 - \frac{4}{5} (3 a + 10) = \frac{1}{10} (4 - 2 a)$
Solve for s: $A = π r^{2} + π r s$
Solve for x: $y = m x + b$
A larger integer is 3 more than twice another. If their sum divided by 2 is 9, find the integers.
The sum of three consecutive odd integers is 171. Find the integers.
The length of a rectangle is 3 meters less than twice its width. If the perimeter measures 66 meters, find the length and width.
How long will it take $500 to earn $124 in simple interest earning 6.2% annual interest?
It took Sally $3 \frac{1}{2}$ hours to drive the 147 miles home from her grandmother’s house. What was her average speed?
Jeannine invested her bonus of $8,300 in two accounts. One account earned $3 \frac{1}{2}$ % simple interest and the other earned $4 \frac{3}{4}$ % simple interest. If her total interest for one year was $341.75, how much did she invest in each account?

Solving Linear Inequalities with One Variable

Solve. Graph all solutions on a number line and provide the corresponding interval notation.

$5 x - 7 < 18$
$2 x - 1 > 2$
$9 - x \leq 3$
$3 - 7 x \geq 10$
$61 - 3 (x + 3) > 13$
$7 - 3 (2 x - 1) \geq 6$
$\frac{1}{3} (9 x + 15) - \frac{1}{2} (6 x - 1) < 0$
$\frac{2}{3} (12 x - 1) + \frac{1}{4} (1 - 32 x) < 0$
$20 + 4 (2 a - 3) \geq \frac{1}{2} a + 2$
$\frac{1}{3} (2 x + \frac{3}{2}) - \frac{1}{4} x < \frac{1}{2} (1 - \frac{1}{2} x)$
$- 4 \leq 3 x + 5 < 11$
$5 < 2 x + 15 \leq 13$
$- 1 < 4 (x + 1) - 1 < 9$
$0 \leq 3 (2 x - 3) + 1 \leq 10$
$- 1 < \frac{2 x - 5}{4} < 1$
$- 2 \leq \frac{3 - x}{3} < 1$
$2 x + 3 < 13 and 4 x - 1 > 10$
$3 x - 1 \leq 8 and 2 x + 5 \geq 23$
$5 x - 3 < - 2 or 5 x - 3 > 2$
$1 - 3 x \leq - 1 or 1 - 3 x \geq 1$
$5 x + 6 < 6 or 9 x - 2 > - 11$
$2 (3 x - 1) < - 16 or 3 (1 - 2 x) < - 15$
Jerry scored 90, 85, 92, and 76 on the first four algebra exams. What must he score on the fifth exam so that his average is at least 80?
If 6 degrees less than 3 times an angle is between 90 degrees and 180 degrees, then what are the bounds of the original angle?

Answers

$\frac{7}{15}$
$\frac{31}{18}$
$\frac{1}{2}$
$- a$
$[- 10, \infty)$ ;
$[- 8, 0)$ ;
$[- 1, 3)$ ;
$ℝ$ ;
$x \geq - 8$
$12 \leq x \leq 32$
$x \leq 1 o r x > 5$
$x > - 4$
$- \frac{3}{4}$
4
$a = \pm 6$
$Ø$

$\frac{1}{5}$
−4
$- \frac{8}{27}$
−15
−6
−24
$- \frac{34}{7}$
−50
14

$6 \sqrt{2}$
0
$\frac{5 \sqrt{3}}{4}$
$2 \sqrt[3]{5}$
$- 3 \sqrt[3]{3}$
$\frac{5 \sqrt[3]{2}}{3}$
3.464
2.621
$8 \sqrt{2}$ centimeters

$6 x^{2} + 2 x - 4$
$- 3 a^{2} + 15 a b + 6 b^{2}$
$x^{2} y - 10 x y^{2}$
$5 a b + 1$
$- x^{2} + 16 x + 3$
$- a^{2} - a b + 2 b^{2}$
$\frac{11}{4}$
80
30
$2 \sqrt{6}$
$60 π$
$855
1.6 seconds

$x^{7}$
$- 21 x^{6} y^{3} z^{4}$
$- \frac{2 a^{7}}{5 b^{2} c}$
$\frac{x^{20} y^{12}}{16 z^{4}}$
$\frac{25 a^{4} b^{6}}{c^{10}}$
$- \frac{27 a^{9}}{8 b^{15} c^{3}}$
$1.333 \times 10^{15}$
$7 \times 10^{- 23}$
$128.57

$- x^{2} - 2 x + 2$
$2 a^{2} b^{2} - 2 a b - 13$
$- 12 x^{2} - 6 x + 3$
$2 x^{2} - 3 x - 20$
$2 x^{4} - 5 x^{3} + 16 x^{2} - 13 x + 20$
$8 a^{3} + b^{3}$
$27 x^{3} - 27 x^{2} + 9 x - 1$
$x^{4} - 2 x^{2} y^{2} + y^{4}$
$9 a + 27 b - 3$
$x^{2} - 3 x + 2$
$5 x^{2} - x + 2 + \frac{5}{x - 4}$
$a^{2} - a + 2$

$\frac{5}{3}$
$\frac{14}{5}$
$\frac{1}{18}$
$\frac{1}{3}$
Ø
−3
$ℝ$
$- \frac{7}{2}$
$s = \frac{A - π r^{2}}{π r}$
5, 13
Length: 21 meters; Width: 12 meters
42 miles per hour

$(- \infty, 5)$ ;
$[6, \infty)$ ;
$(- \infty, 13)$ ;
$Ø$ ;
$[- \frac{4}{5}, \infty)$ ;
$[- 3, 2)$ ;
$(- 1, \frac{3}{2})$ ;
$(\frac{1}{2}, \frac{9}{2})$ ;
$(\frac{11}{4}, 5)$ ;
$(- \infty, \frac{1}{5}) \cup (1, \infty)$ ;
$ℝ$ ;
Jerry must score at least 57 on the fifth exam.

Sample Exam

Simplify.

$5 - 3 (12 - | 2 - 5^{2} |)$
${(- \frac{1}{2})}^{2} - {(3 - 2 | - \frac{3}{4} |)}^{3}$
$- 7 \sqrt{60}$
$5 \sqrt[3]{- 32}$
Find the diagonal of a square with sides measuring 6 centimeters.

Simplify.

$- 5 x^{2} y z^{- 1} (3 x^{3} y^{- 2} z)$
${(\frac{- 2 a^{- 4} b^{2} c}{a^{- 3} b^{0} c^{2}})}^{- 3}$
$2 (3 a^{2} b^{2} + 2 a b - 1) - a^{2} b^{2} + 2 a b - 1$
$(x^{2} - 6 x + 9) - (3 x^{2} - 7 x + 2)$
${(2 x - 3)}^{3}$
$(3 a - b) (9 a^{2} + 3 a b + b^{2})$
$\frac{6 x^{4} - 17 x^{3} + 16 x^{2} - 18 x + 13}{2 x - 3}$

Solve.

$\frac{4}{5} x - \frac{2}{15} = 2$
$\frac{3}{4} (8 x - 12) - \frac{1}{2} (2 x - 10) = 16$
$12 - 5 (3 x - 1) = 2 (4 x + 3)$
$\frac{1}{2} (12 x - 2) + 5 = 4 (\frac{3}{2} x - 8)$
Solve for y: $a x + b y = c$

Solve. Graph the solutions on a number line and give the corresponding interval notation.

$2 (3 x - 5) - (7 x - 3) \geq 0$
$2 (4 x - 1) - 4 (5 + 2 x) < - 10$
$- 6 \leq \frac{1}{4} (2 x - 8) < 4$
$3 x - 7 > 14 or 3 x - 7 < - 14$

Use algebra to solve the following.

Degrees Fahrenheit F is given by the formula $F = \frac{9}{5} C + 32$ where C represents degrees Celsius. What is the Fahrenheit equivalent to 35° Celsius?
The length of a rectangle is 5 inches less than its width. If the perimeter is 134 inches, find the length and width of the rectangle.
Melanie invested 4,500 in two separate accounts. She invested part in a CD that earned 3.2% simple interest and the rest in a savings account that earned 2.8% simple interest. If the total simple interest for one year was $138.80, how much did she invest in each account?
A rental car costs $45.00 per day plus $0.48 per mile driven. If the total cost of a one-day rental is to be at most $105, how many miles can be driven?

Answers

38
$- 14 \sqrt{15}$
$6 \sqrt{2}$ centimeters
$- \frac{a^{3} c^{3}}{8 b^{6}}$
$- 2 x^{2} + x + 7$
$27 a^{3} - b^{3}$
$\frac{8}{3}$
$\frac{11}{23}$
$y = \frac{c - a x}{b}$
$ℝ$ ;
$(- \infty, - \frac{7}{3}) \cup (7, \infty)$ ;
Length: 31 inches; width: 36 inches
The car can be driven at most 125 miles.