This review covers every major topic from Pre-Algebra in one place: variables, expressions, equations, inequalities, exponents, proportions, absolute value, and graphing. Each section has a quick refresher and a worked example so you can brush up before moving on.
A variable is a letter that stands for an unknown number. An expression combines variables, numbers, and operations (but no equals sign).
To evaluate an expression, substitute the given values and follow order of operations.
Evaluate 3 x + 2 y − 5 3x + 2y - 5 3 x + 2 y − 5 x = 4 x = 4 x = 4 y = 3 y = 3 y = 3
3 ( 4 ) + 2 ( 3 ) − 5 = 12 + 6 − 5 = 13 \begin{aligned}
3(4) + 2(3) - 5 &\\
= 12 + 6 - 5 &\\
= 13
\end{aligned} 3 ( 4 ) + 2 ( 3 ) − 5 = 12 + 6 − 5 = 13 Same signs → add and keep the sign. Different signs → subtract and keep the sign of the larger absolute value.
For multiplication/division: same signs → positive, different signs → negative.
− 8 + 5 = − 3 ( − 6 ) × ( − 4 ) = 24 15 ÷ ( − 3 ) = − 5 \begin{aligned}
{-}8 + 5 &= {-}3 \\
({-}6) \times ({-}4) &= 24 \\
15 \div ({-}3) &= {-}5
\end{aligned} − 8 + 5 ( − 6 ) × ( − 4 ) 15 ÷ ( − 3 ) = − 3 = 24 = − 5 PEMDAS: Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right).
Evaluate 4 + 3 ( 2 x − 1 ) 2 4 + 3(2x - 1)^2 4 + 3 ( 2 x − 1 ) 2 x = 2 x = 2 x = 2
4 + 3 ( 2 ( 2 ) − 1 ) 2 = 4 + 3 ( 4 − 1 ) 2 = 4 + 3 ( 3 ) 2 = 4 + 3 ( 9 ) = 4 + 27 = 31 \begin{aligned}
4 + 3(2(2) - 1)^2 &\\
= 4 + 3(4 - 1)^2 &\\
= 4 + 3(3)^2 &\\
= 4 + 3(9) &\\
= 4 + 27 = 31
\end{aligned} 4 + 3 ( 2 ( 2 ) − 1 ) 2 = 4 + 3 ( 4 − 1 ) 2 = 4 + 3 ( 3 ) 2 = 4 + 3 ( 9 ) = 4 + 27 = 31 Like terms have the same variable and exponent. Combine by adding/subtracting coefficients.
The distributive property : a (b + c ) = ab + ac .
Simplify 2 ( 3 x + 4 ) − 5 x + 1 2(3x + 4) - 5x + 1 2 ( 3 x + 4 ) − 5 x + 1
2 ( 3 x + 4 ) − 5 x + 1 = 6 x + 8 − 5 x + 1 = x + 9 \begin{aligned}
2(3x + 4) - 5x + 1 &\\
= 6x + 8 - 5x + 1 &\\
= x + 9
\end{aligned} 2 ( 3 x + 4 ) − 5 x + 1 = 6 x + 8 − 5 x + 1 = x + 9 One-step : Do the inverse operation on both sides.
x + 12 = 20 x = 8 \begin{aligned}
x + 12 &= 20 \\
x &= 8
\end{aligned} x + 12 x = 20 = 8 Two-step : Undo add/subtract first, then multiply/divide.
3 x − 7 = 14 3 x = 21 x = 7 \begin{aligned}
3x - 7 &= 14 \\
3x &= 21 \\
x &= 7
\end{aligned} 3 x − 7 3 x x = 14 = 21 = 7 Variables on both sides : Move all variable terms to one side first.
5 x + 3 = 2 x + 18 3 x + 3 = 18 3 x = 15 x = 5 \begin{aligned}
5x + 3 &= 2x + 18 \\
3x + 3 &= 18 \\
3x &= 15 \\
x &= 5
\end{aligned} 5 x + 3 3 x + 3 3 x x = 2 x + 18 = 18 = 15 = 5 Check: 5(5) + 3 = 28 and 2(5) + 18 = 28 ✓
Solve like equations, but flip the sign when multiplying/dividing by a negative.
− 2 x + 6 > 12 − 2 x > 6 x < − 3 (flip: divided by − 2 ) \begin{aligned}
{-}2x + 6 &> 12 \\
{-}2x &> 6 \\
x &< {-}3 \quad \text{(flip: divided by }{-}2\text{)}
\end{aligned} − 2 x + 6 − 2 x x > 12 > 6 < − 3 (flip: divided by − 2 ) Graph: open circle at −3, shade left.
a n a^n a n a by itself n times. Key rules:
a 0 = 1 a^0 = 1 a 0 = 1 a − n = 1 a n a^{-n} = \frac{1}{a^n} a − n = a n 1 a m × a n = a m + n a^m \times a^n = a^{m+n} a m × a n = a m + n
2 5 = 32 4 − 2 = 1 16 3 2 × 3 3 = 3 5 = 243 \begin{aligned}
2^5 &= 32 \\
4^{-2} &= \frac{1}{16} \\
3^2 \times 3^3 &= 3^5 = 243
\end{aligned} 2 5 4 − 2 3 2 × 3 3 = 32 = 16 1 = 3 5 = 243 A proportion is two equal ratios: a /b = c /d . Cross-multiply to solve.
A recipe uses 3 cups of flour for 18 muffins. How much for 30 muffins?
3 18 = x 30 3 × 30 = 18 x 90 = 18 x x = 5 cups \begin{aligned}
\frac{3}{18} &= \frac{x}{30} \\
3 \times 30 &= 18x \\
90 &= 18x \\
x &= 5 \text{ cups}
\end{aligned} 18 3 3 × 30 90 x = 30 x = 18 x = 18 x = 5 cups ∣ x ∣ |x| ∣ x ∣
Solve ∣ 2 x − 1 ∣ = 7 |2x - 1| = 7 ∣2 x − 1∣ = 7
Case 1:
2 x − 1 = 7 2 x = 8 x = 4 \begin{aligned}
2x - 1 &= 7 \\
2x &= 8 \\
x &= 4
\end{aligned} 2 x − 1 2 x x = 7 = 8 = 4 Case 2:
2 x − 1 = − 7 2 x = − 6 x = − 3 \begin{aligned}
2x - 1 &= {-}7 \\
2x &= {-}6 \\
x &= {-}3
\end{aligned} 2 x − 1 2 x x = − 7 = − 6 = − 3 Check: |2(4) − 1| = |7| = 7 ✓ and |2(−3) − 1| = |−7| = 7 ✓
Translate words into equations, solve, and check.
You earn 18 dollars per hour and need at least 500 dollars for rent. How many hours minimum?
18 h ≥ 500 h ≥ 27.8 \begin{aligned}
18h &\geq 500 \\
h &\geq 27.8
\end{aligned} 18 h h ≥ 500 ≥ 27.8 You need at least 28 hours.
Points are (x , y ). Lines in slope-intercept form: y = mx + b .
Graph y = 2 x − 1 y = 2x - 1 y = 2 x − 1
x x x y y y 0 0 0 − 1 -1 − 1 1 1 1 1 1 1 2 2 2 3 3 3 − 1 -1 − 1 − 3 -3 − 3
y = 2x - 1 x y -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 y = 2x - 1 (-1, -3) (0, -1) (1, 1) (2, 3)
Slope m = 2 (up 2, right 1). Y-intercept b = −1 (crosses y-axis at (0, −1)).
Try these on your own before checking the quiz:
Evaluate 4 ( x − 3 ) + 2 x 4(x - 3) + 2x 4 ( x − 3 ) + 2 x x = 5 x = 5 x = 5 4 ( 2 ) + 10 = 18 4(2) + 10 = 18 4 ( 2 ) + 10 = 18
Solve 6 x + 1 = 3 x + 16 6x + 1 = 3x + 16 6 x + 1 = 3 x + 16 x = 5 x = 5 x = 5
Solve − 3 y ≤ 21 -3y \leq 21 − 3 y ≤ 21 y ≥ − 7 y \geq -7 y ≥ − 7
Simplify 5 3 × 5 − 1 5^3 \times 5^{-1} 5 3 × 5 − 1 5 2 = 25 5^2 = 25 5 2 = 25
Solve 4 7 = x 21 \frac{4}{7} = \frac{x}{21} 7 4 = 21 x x = 12 x = 12 x = 12
Solve ∣ x − 4 ∣ = 9 |x - 4| = 9 ∣ x − 4∣ = 9 x = 13 x = 13 x = 13 x = − 5 x = -5 x = − 5
You’ve covered a lot in Pre-Algebra, from letters standing for numbers to solving equations, inequalities, and even graphing. Each skill builds on the last. Pick one problem type at a time, check your work, and celebrate what you’ve already mastered. You’re ready for Algebra Basics whenever you are.
Evaluate $5 + 2(3x - 1)$ when $x = 4$.
Solve $7x - 9 = 4x + 6$. A. $x = 5$ B. $x = 3$ C. $x = -5$ D. $x = 1$
What is $(-9) \times (-7)$? A. $63$ B. $-63$ C. $16$ D. $-16$
Simplify $2^4 \times 2^{-1}$. A. $2^3 = 8$ B. $2^5 = 32$ C. $2^{-4}$ D. $16$
Solve $-5x > 20$. A. $x < -4$ B. $x > -4$ C. $x < 4$ D. $x > 4$
Solve $|x + 2| = 8$. A. $x = 6$ or $x = -10$ B. $x = 6$ only C. $x = -6$ only D. No solution
A trip is 150 miles at 50 mph. How long? A. 3 hours B. 2 hours C. 4 hours D. 2.5 hours
What is the slope of $y = -4x + 7$?
Simplify $4(3x - 2) + 5x$. A. $17x - 8$ B. $12x - 8$ C. $17x + 8$ D. $7x - 8$
Solve $\frac{x}{5} + 3 = 7$. A. $x = 20$ B. $x = 2$ C. $x = 50$ D. $x = 35$
What is $(-4)^3$? A. $-64$ B. $64$ C. $-12$ D. $12$
Solve $|2x - 6| = 10$. A. $x = 8$ or $x = -2$ B. $x = 8$ only C. $x = -2$ only D. No solution
In which quadrant is the point $(-3, -5)$? A. Quadrant III B. Quadrant I C. Quadrant II D. Quadrant IV
Solve $\frac{4}{x} = \frac{8}{6}$. A. $x = 3$ B. $x = 12$ C. $x = 2$ D. $x = 6$
Simplify $6x - 2(x + 4)$. A. $4x - 8$ B. $4x + 8$ C. $8x - 8$ D. $4x - 4$
Solve $3(x + 2) = 2(x + 5)$. A. $x = 4$ B. $x = 2$ C. $x = 8$ D. $x = 16$
What is $\frac{2^5}{2^2}$? A. $2^3 = 8$ B. $2^7 = 128$ C. $2^{10}$ D. $4$
Solve $-2x + 5 \leq 11$. A. $x \geq -3$ B. $x \leq -3$ C. $x \geq 3$ D. $x \leq 3$
Evaluate $3x^2 - x + 2$ when $x = -1$.
A recipe needs $\frac{3}{4}$ cup of sugar per batch. How much for $2\frac{1}{2}$ batches? A. $1\frac{7}{8}$ cups B. $2$ cups C. $1\frac{1}{2}$ cups D. $2\frac{1}{4}$ cups
Retry Quiz
