Chapter 1: Introduction to Real Numbers and Algebraic
Expressions
Evaluate $\frac{-x}{-y}$ for $x = 72$ and $y = 8$.
$9$
Evaluate $x + y$ for $x = 38$ and $y = 62$.
$100$
Translate "Six more than some number" into an algebraic
expression.
$x + 6$
Translate "One-eighth of a number" into an algebraic
expression.
$\frac{1}{8}y$ or $\frac{y}{8}$
Perform: $12 + (-9)$.
$3$
Perform: $-8 + 5$.
$-3$
Subtract: $3 - 7$.
$-4$
Subtract: $-4 - (-10)$.
$6$
Simplify: $7(x - y)$.
$7x - 7y$
Simplify: $-(4x + 5y + 2)$.
$-4x - 5y - 2$
Chapter 2: Solving Equations and Inequalities
Solve for $x$: $4x + 7 - 6x = 10 + 3x + 12$.
$x = -3$
Solve for $x$: $3 - 8(x + 6) = 4(x - 1) - 5$.
$x = -3$
Solve $d = rt$ for $r$.
$r = \frac{d}{t}$
Solve $C = \pi d$ for $d$.
$d = \frac{C}{\pi}$
Solve: "What is 25% of 60?"
$a = 0.25 \times 60$; $a = 15$
A 480-inch pipe is cut into two pieces. One piece is
three times the length of the other. Find the length of
each piece.
One piece is 120 inches, and the other is 360 inches.
Determine whether -4 is a solution to the inequality $x
< 5$.
Yes, because $-4 < 5$ is true.
Solve: $7x \leq 35$.
$x \leq 5$
Solve: $-4y < 20$.
$y > -5$ (Note: The inequality symbol reverses when
dividing by a negative number).
Solve: $3x - 3 > x + 7$.
$x > 5$
Chapter 3: Graphs of Linear Equations
Determine the quadrant of the point $(-4, 5)$.
Quadrant II
Determine the quadrant of the point $(3, -4)$.
Quadrant IV
Determine if $(2, 3)$ is a solution to $4y + 3x = 18$.
Yes ($4(3) + 3(2) = 12 + 6 = 18$).
Identify the origin.
$(0, 0)$
Find the $y$-intercept of $5x + 2y = 10$.
$(0, 5)$
Find the $x$-intercept of $5x + 2y = 10$.
$(2, 0)$
Describe the graph of $y = 2$.
A horizontal line passing through $(0, 2)$ with a slope
of 0.
Describe the graph of $x = -2$.
A vertical line passing through $(-2, 0)$ with an
undefined slope.
Find the slope through $(-4, 5)$ and $(4, -1)$.
$m = -\frac{3}{4}$
Find the slope of $3x + 5y = 15$.
$m = -\frac{3}{5}$
Chapter 4: Polynomials and Operations
Evaluate $(6x)^3$ when $x = -3$.
$-5832$
Evaluate $6x^3$ when $x = -3$.
$-162$
Convert $68,000$ to scientific notation.
$6.8 \times 10^4$
Convert $0.000567$ to scientific notation.
$5.67 \times 10^{-4}$
Identify the degree: $4x^2 - 9x^3 + 6x^4 + 8x - 7$.
4 (The highest exponent).
Combine like terms: $7x^5 + 9 + 3x^2 + 6x^2 - 13 -
6x^5$.
$x^5 + 9x^2 - 4$
Add: $(-6x^3 + 7x - 2) + (5x^3 + 4x^2 + 3)$.
$-x^3 + 4x^2 + 7x + 1$
Subtract: $(5x^2y + 2x^3y^2 + 4x^2y^3 + 7y) - (5x^2y -
7x^3y^2 + x^2y^2 - 6y)$.
$9x^3y^2 + 4x^2y^3 - x^2y^2 + 13y$
Multiply: $(x + 4)(x^2 + 3)$.
$x^3 + 4x^2 + 3x + 12$
Divide: $(21a^5b^4 - 14a^3b^2 + 7a^2b) \div (-7a^2b)$.
$-3a^3b^3 + 2ab - 1$
Chapter 5: Polynomials: Factoring
Find the GCF of $30x^3, -48x^4, 54x^5, 12x^2$.
$6x^2$
Factor: $28x^6 + 32x^3$.
$4x^3(7x^3 + 8)$
Factor: $3x^3 + 9x^2 + x + 3$.
$(3x^2 + 1)(x + 3)$
Factor: $x^2 + 8x + 16$.
$(x + 4)^2$
Factor: $x^2 - 9$.
$(x + 3)(x - 3)$
Factor: $y^2 - 8y + 15$.
$(y - 3)(y - 5)$
Factor: $4x^2 - 5x - 6$.
$(x - 2)(4x + 3)$
Solve: $(x + 4)(x - 3) = 0$.
$x = -4$ or $x = 3$
Solve: $x^2 + 9x + 14 = 0$.
$x = -7$ or $x = -2$
Chapter 6: Rational Expressions and Equations
Find excluded values for $\frac{x+3}{x^2 - 3x - 28}$.
$x = 7$ and $x = -4$
Simplify: $\frac{3x - 12}{5x - 20}$.
$\frac{3}{5}$
Multiply: $\frac{6a^4}{10} \cdot \frac{5}{6a}$.
$\frac{a^3}{2}$
Divide: $\frac{x}{9} \div \frac{8}{y}$.
$\frac{xy}{72}$
Find the LCM of $15x, 30y, 25xyz$.
$150xyz$
Add: $\frac{4x}{x - 7} + \frac{3x + 2}{x - 7}$.
$\frac{7x + 2}{x - 7}$
Subtract: $\frac{5x}{x + 3} - \frac{x - 4}{x + 3}$.
$\frac{4x + 4}{x + 3}$
Solve: $\frac{x}{5} = \frac{x}{2} - \frac{1}{4}$.
$x = \frac{5}{6}$
Solve: $\frac{1}{3x} + \frac{1}{x} = \frac{4}{15}$.
$x = 5$
Solve for $x$: $\frac{x-2}{x-3} = \frac{x-1}{x+1}$.
$x = \frac{5}{3}$
Chapter 7: Graphs, Functions, and Applications
Determine if the correspondence $\{(6, 1), (-8, 3), (19,
1)\}$ is a function.
Yes (Each domain member corresponds to exactly one range
member).
Find $f(-2)$ for $f(x) = 3x^2 + 2x$.
$8$
Determine the domain of $f(x) = \frac{3x}{x - 4}$.
All real numbers except 4 (or $\{x | x \neq 4\}$).
Find the $y$-intercept of $f(x) = -3.1x + 7$.
$(0, 7)$
Find slope and $y$-intercept of $4x - 7f(x) = 2$.
Slope = $\frac{4}{7}$, y-intercept = $(0, -\frac{2}{7})$
Find equation with slope $3$ through $(2, 7)$.
$y = 3x + 1$
Find equation through $(2, 2)$ and $(-6, -4)$.
$y = \frac{3}{4}x + \frac{1}{2}$
Find equation parallel to $y = -3x + 4$ through $(1,
-5)$.
$y = -3x - 2$
Find equation perpendicular to $7x - 2y = -2$ through
$(7, 1)$.
$y = -\frac{2}{7}x + 3$
Determine if $y = 2x + 3$ and $y = 2x - 1$ are parallel,
perpendicular, or neither.
Parallel (same slope, different y-intercepts).
Chapter 8: Systems of Equations
Solve: $$y = 3x + 1$$ $$x - 2y = 3$$
$(-1, -2)$
Determine if consistent or inconsistent: $$y =
\frac{3}{4}x + 2$$ $$y = \frac{3}{4}x - 3$$
Inconsistent (Parallel lines have no intersection).
Determine if dependent or independent: $$4x + 8y = 16$$
$$2x + 4y = 8$$
Dependent (They represent the same line).
Solve: $$x = 3 - y$$ $$5x + 3y = 5$$
$(-2, 5)$
Solve: $$2x + y = 5$$ $$x - y = 4$$
$(3, -1)$
Solve: $$4x + 3y = 8$$ $$x - 3y = 7$$
$(3, -\frac{4}{3})$
Solve: $$2x + 5y = -1$$ $$3x - 10y = 16$$
$(2, -1)$
If solving a system results in the statement $0 = -4$,
what is the solution set?
No solution (Inconsistent system).
If solving a system results in the statement $0 = 0$,
what does this indicate?
Infinite solutions (Dependent equations).
Classify the system: One solution found (lines
intersect).
Consistent and Independent.
Chapter 9: More on Inequalities
Determine if 6 is a solution to $3y + 2 > 7$.
Yes ($20 > 7$ is true).
Write interval notation for the set $\{x | -1 < x <
9\}$.
$(-1, 9)$
Solve: $3x - 3 > x + 7$.
$x > 5$; Graph shows an open parenthesis at 5 shaded to
the right.
Solve: $2x + 1 \geq -3$ AND $3x < 12$.
$[-2, 4)$
Simplify: $|-8y|$.
$8|y|$
Solve: $|2x + 1| = 5$.
$x = 2$ or $x = -3$
Solve: $|3x - 5| = |8 + 4x|$.
$x = -13$ or $x = -\frac{3}{7}$
Solve: $|x| < 3$.
$(-3, 3)$
Solve: $|x| \geq 3$.
$(-\infty, -3] \cup [3, \infty)$
Determine if $(5, 2)$ is a solution to $3x - 2y > 12$.
No ($11 > 12$ is false).