What is the value of \(|-7|\)?Concept: Absolute value = distance from zero. Always non-negative. Since \(|-7| = 7\) (distance from 0 to −7 on the number line), the answer is 7. Which property is shown: \(3 + 5 = 5 + 3\)?Concept: Number properties: Commutative (order), Associative (grouping), Distributive (expand), Identity (add 0 or multiply by 1). The Commutative Property of Addition states \(a+b = b+a\). Order changed, but sum is the same. Simplify: \(-3 + (-4)\)Concept: Adding two negatives = add the values, keep the negative sign. \(-3 + (-4) = -(3+4) = \mathbf{-7}\). Simplify: \(-8 - (-3)\)Concept: Subtracting a negative = adding a positive: \(a - (-b) = a + b\). \(-8 - (-3) = -8 + 3 = \mathbf{-5}\). What is \((-2)(-6)\)?Concept: Negative × Negative = Positive. \((-2)(-6) = +12\). Two negatives multiply to a positive. What is \(-18 \div 3\)?Concept: Negative ÷ Positive = Negative. \(-18 \div 3 = \mathbf{-6}\). Different signs → negative result. Which is the greatest: \(-1,\ -5,\ 0,\ -3\)?Concept: On the number line, farther right = greater. Zero is greater than all negatives. \(0 > -1 > -3 > -5\). The greatest value is 0. Simplify: \(2^3\)Concept: \(2^3\) means \(2 \times 2 \times 2\). \(2^3 = 2 \times 2 \times 2 = \mathbf{8}\). What is \(\sqrt{49}\)?Concept: The square root asks: what number times itself equals 49? \(7 \times 7 = 49\), so \(\sqrt{49} = \mathbf{7}\). Find the LCM of 4 and 6.Concept: LCM = Least Common Multiple = smallest number both divide evenly into. Multiples of 4: 4,8,12,... Multiples of 6: 6,12,... LCM = 12. Find the GCF of 12 and 18.Concept: GCF = Greatest Common Factor = largest number that divides both. Factors of 12: 1,2,3,4,6,12. Factors of 18: 1,2,3,6,9,18. GCF = 6. Convert \(0.75\) to a fraction in lowest terms.Concept: \(0.75 = \frac{75}{100}\). Then simplify by dividing by GCF. \(\frac{75}{100} \div \frac{25}{25} = \dfrac{3}{4}\). Add: \(\dfrac{1}{2} + \dfrac{1}{3}\)Concept: Find a common denominator first. LCD of 2 and 3 is 6. \(\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\). Subtract: \(\dfrac{3}{4} - \dfrac{1}{6}\)Concept: LCD of 4 and 6 is 12. \(\dfrac{9}{12} - \dfrac{2}{12} = \dfrac{7}{12}\). Multiply: \(\dfrac{2}{3} \times \dfrac{3}{4}\)Concept: Multiply numerators together, denominators together. Then simplify. \(\dfrac{2 \times 3}{3 \times 4} = \dfrac{6}{12} = \dfrac{1}{2}\). Divide: \(\dfrac{3}{4} \div \dfrac{1}{2}\)Concept: Dividing by a fraction = multiplying by its reciprocal. \(\dfrac{3}{4} \times \dfrac{2}{1} = \dfrac{6}{4} = \dfrac{3}{2}\). Convert \(\dfrac{3}{5}\) to a decimal.Concept: Divide the numerator by the denominator. \(3 \div 5 = \mathbf{0.6}\). Order from least to greatest: \(\dfrac{1}{4},\ \dfrac{1}{3},\ \dfrac{1}{5}\)Concept: Convert to decimals or common denominator to compare. \(\frac{1}{5}=0.2,\ \frac{1}{4}=0.25,\ \frac{1}{3} \approx 0.33\). Order: \(\dfrac{1}{5},\ \dfrac{1}{4},\ \dfrac{1}{3}\). Multiply: \(1.2 \times 0.5\)Concept: Multiply as whole numbers, then count total decimal places (2 total). \(12 \times 5 = 60\). Two decimal places → \(\mathbf{0.60} = 0.6\). Divide: \(2.4 \div 0.4\)Concept: Multiply both by 10 to remove decimals: \(24 \div 4\). \(24 \div 4 = \mathbf{6}\). What is 25% of 80?Concept: "Percent of" means multiply. 25% = 0.25. \(0.25 \times 80 = \mathbf{20}\). Convert \(2\frac{1}{3}\) to an improper fraction.Concept: Multiply whole number by denominator, add numerator: \((2 \times 3 + 1) / 3\). \(\frac{6+1}{3} = \dfrac{7}{3}\). Simplify the ratio \(8:12\).Concept: Divide both parts by their GCF. GCF(8,12) = 4. \(8\div4 : 12\div4 = \mathbf{2:3}\). Solve for \(x\): \(\dfrac{3}{x} = \dfrac{6}{10}\)Concept: Cross-multiply: \(3 \times 10 = 6 \times x\). \(30 = 6x \Rightarrow x = 5\). 40 is what percent of 200?Concept: \(\text{percent} = \dfrac{\text{part}}{\text{whole}} \times 100\). \(\dfrac{40}{200} \times 100 = 20\%\). Find 15% of 60.Concept: 15% = 0.15. Multiply by the whole. \(0.15 \times 60 = \mathbf{9}\). A map uses a scale of 1 in = 50 mi. What real distance does 3 inches represent?Concept: Multiply map distance by the scale factor. \(3 \times 50 = \mathbf{150}\) miles. A \$40 item is discounted by 10%. What is the sale price?Concept: Sale price = original × (1 − discount%). Here 1 − 0.10 = 0.90. \(40 \times 0.90 = \mathbf{\$36}\). The ratio of boys to girls is 2:3. If there are 25 students total, how many are girls?Concept: Girls are \(\frac{3}{5}\) of the total (3 parts out of 2+3=5). \(25 \times \dfrac{3}{5} = \mathbf{15}\) girls. If 5 apples cost \$2, what do 15 apples cost?Concept: Unit rate: cost per apple = \(\frac{2}{5}\). Multiply by 15. \(\dfrac{2}{5} \times 15 = \mathbf{\$6}\). A price increases from \$50 to \$75. What is the percent increase?Concept: \(\text{%increase} = \dfrac{\text{new} - \text{old}}{\text{old}} \times 100\). \(\dfrac{75-50}{50} \times 100 = \dfrac{25}{50} \times 100 = \mathbf{50\%}\). Express \(0.35\) as a percent.Concept: Multiply by 100 (move decimal two places right). \(0.35 \times 100 = \mathbf{35\%}\). A \$25 item has 8% sales tax. What is the total cost?Concept: Total = price × (1 + tax%). Here \(1.08 \times 25\). \(25 \times 1.08 = \mathbf{\$27}\). Evaluate \(3x + 2\) when \(x = 4\).Concept: Substitute the value of \(x\), then compute. \(3(4) + 2 = 12 + 2 = \mathbf{14}\). Simplify: \(2x + 3x\)Concept: Combine like terms — add coefficients (numbers in front of same variable). \(2x + 3x = (2+3)x = \mathbf{5x}\). Expand: \(3(x + 4)\)Concept: Distributive Property: \(a(b+c) = ab + ac\). \(3 \cdot x + 3 \cdot 4 = \mathbf{3x + 12}\). Simplify: \(4x + 2 - x + 3\)Concept: Group and combine like terms (\(x\)-terms together, constants together). \((4x - x) + (2 + 3) = \mathbf{3x + 5}\). Evaluate \(x^2 - 2x + 1\) when \(x = 3\).Concept: Substitute \(x=3\) and follow order of operations (exponents first). \(9 - 6 + 1 = \mathbf{4}\). What is the coefficient in the term \(7y\)?Concept: Coefficient = the numerical part multiplied by the variable. In \(7y\), the coefficient is 7. Identify the constant in: \(5x + 3y - 8\)Concept: A constant is a number with no variable attached. The constant term is \(\mathbf{-8}\) (no variable). Simplify: \(2(3x - 1) + 4\)Concept: Distribute first, then combine like terms. \(6x - 2 + 4 = \mathbf{6x + 2}\). Combine: \(5a - 2b + 3a + b\)Concept: Combine \(a\)-terms and \(b\)-terms separately. \((5a+3a) + (-2b+b) = \mathbf{8a - b}\). Evaluate \(2a + b\) when \(a = 3,\ b = -1\).Concept: Substitute both values, then compute. \(2(3) + (-1) = 6 - 1 = \mathbf{5}\). Factor out the GCF from \(6x + 9\).Concept: GCF of 6 and 9 is 3. Factor it out. \(6x + 9 = \mathbf{3(2x + 3)}\). Check: \(3\cdot2x + 3\cdot3 = 6x+9\). Solve: \(x + 5 = 12\)Concept: To isolate \(x\), subtract 5 from both sides. \(x = 12 - 5 = \mathbf{7}\). Solve: \(3x = 21\)Concept: Divide both sides by 3. \(x = 21 \div 3 = \mathbf{7}\). Solve: \(x - 4 = 9\)Concept: Add 4 to both sides. \(x = 9 + 4 = \mathbf{13}\). Solve: \(\dfrac{x}{2} = 8\)Concept: Multiply both sides by 2. \(x = 8 \times 2 = \mathbf{16}\). Solve: \(2x + 3 = 11\)Concept: Step 1: subtract 3. Step 2: divide by 2. \(2x = 8 \Rightarrow x = \mathbf{4}\). Solve: \(4x - 7 = 13\)Concept: Add 7 to both sides, then divide by 4. \(4x = 20 \Rightarrow x = \mathbf{5}\). Solve: \(3x + 2 = 2x + 9\)Concept: Move all \(x\)-terms to one side, constants to the other. \(3x - 2x = 9 - 2 \Rightarrow x = \mathbf{7}\). Solve: \(-2x = 10\)Concept: Divide both sides by −2. Sign of \(x\) changes. \(x = 10 \div (-2) = \mathbf{-5}\). Solve: \(\dfrac{x}{3} + 1 = 4\)Concept: Subtract 1, then multiply by 3. \(\dfrac{x}{3} = 3 \Rightarrow x = \mathbf{9}\). Solve: \(5(x - 2) = 15\)Concept: Divide both sides by 5 first (or distribute first). \(x - 2 = 3 \Rightarrow x = \mathbf{5}\). Solve: \(2(x + 3) = 3x - 1\)Concept: Distribute left side, then collect \(x\)-terms. \(2x + 6 = 3x - 1 \Rightarrow 7 = x \Rightarrow x = \mathbf{7}\). Two consecutive integers sum to 20. Find the smaller one. (\(x + (x+2) = 20\))Concept: Set up equation: \(x + x + 2 = 20\). \(2x + 2 = 20 \Rightarrow 2x = 18 \Rightarrow x = \mathbf{9}\). Solve: \(\dfrac{3x}{4} = 9\)Concept: Multiply both sides by \(\frac{4}{3}\) (or multiply by 4, then divide by 3). \(3x = 36 \Rightarrow x = \mathbf{12}\). Solve the inequality: \(x + 3 > 7\)Concept: Subtract 3 from both sides (inequality direction stays the same). \(x > 7 - 3 \Rightarrow x > \mathbf{4}\). Solve: \(2x \leq 10\)Concept: Divide both sides by 2. (Positive divisor — direction unchanged.) \(x \leq 5\). Solve: \(-x < 5\)Concept: Multiply/divide by negative → flip the inequality sign. \(x > -5\). Solve: \(3x - 1 \geq 8\)Concept: Add 1 to both sides, then divide by 3. \(3x \geq 9 \Rightarrow x \geq 3\). Which value satisfies \(x > -2\)? Choices: \(-5,\ -3,\ -2,\ 0\)Concept: Test each value — which is strictly greater than −2? \(0 > -2\) ✓. Only 0 satisfies the inequality. The graph of \(x < 3\) on a number line uses what kind of circle at 3?Concept: Open circle = value not included (< or >). Closed = included (≤ or ≥). Since \(x < 3\) does not include 3 itself, use an open circle. Solve: \(2x + 4 < 12\)Concept: Subtract 4, then divide by 2. (Positive — no flip.) \(2x < 8 \Rightarrow x < \mathbf{4}\). Solve: \(-3x \geq 9\)Concept: Dividing by negative −3 → flip inequality sign. \(x \leq -3\). Find the slope of the line through \((0, 0)\) and \((2, 4)\).Concept: \(m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}\). \(m = \dfrac{4-0}{2-0} = \dfrac{4}{2} = \mathbf{2}\). What is the \(y\)-intercept of \(y = 3x + 5\)?Concept: In \(y = mx + b\), \(b\) is the \(y\)-intercept (where the line crosses the y-axis). \(b = \mathbf{5}\). The line crosses the y-axis at \((0, 5)\). What is the slope of \(y = -2x + 1\)?Concept: In slope-intercept form \(y=mx+b\), \(m\) is the slope. The slope is \(m = \mathbf{-2}\). Does the point \((1, 3)\) lie on the line \(y = 2x + 1\)?Concept: Substitute \(x=1\) into the equation. If result equals \(y=3\), it lies on the line. \(y = 2(1)+1 = 3\). Yes, \((1,3)\) lies on the line. Write the equation of a line with slope 2 and \(y\)-intercept −3.Concept: Use slope-intercept form: \(y = mx + b\). \(m = 2,\ b = -3 \Rightarrow y = \mathbf{2x - 3}\). Find the \(x\)-intercept of \(y = 4x - 8\).Concept: Set \(y = 0\) and solve for \(x\). \(0 = 4x - 8 \Rightarrow 4x = 8 \Rightarrow x = \mathbf{2}\). Find the slope through \((1, 2)\) and \((3, 6)\).Concept: \(m = \dfrac{y_2-y_1}{x_2-x_1}\). \(m = \dfrac{6-2}{3-1} = \dfrac{4}{2} = \mathbf{2}\). A table shows: \(x=0 \to y=1\) and \(x=1 \to y=3\). What is the slope?Concept: Slope = change in y ÷ change in x between any two table rows. \(m = \dfrac{3-1}{1-0} = \dfrac{2}{1} = \mathbf{2}\). Two lines are parallel. What must be true about their slopes?Concept: Parallel lines never intersect. They have the same slope but different y-intercepts. Parallel lines have equal slopes. What is the slope of the horizontal line \(y = 4\)?Concept: Horizontal lines have zero rise for any run. Slope = rise/run = 0. Slope = \(\mathbf{0}\). (Vertical lines are undefined.) Find the perimeter of a rectangle with length 8 and width 5.Concept: \(P = 2(l + w)\). \(P = 2(8+5) = 2(13) = \mathbf{26}\) units. Find the area of a rectangle with length 8 and width 5.Concept: \(A = l \times w\). \(A = 8 \times 5 = \mathbf{40}\) sq. units. Find the area of a triangle with base 6 and height 4.Concept: \(A = \dfrac{1}{2} \times b \times h\). \(A = \dfrac{1}{2} \times 6 \times 4 = \mathbf{12}\) sq. units. Find the area of a circle with radius 3. (Use \(\pi \approx 3.14\))Concept: \(A = \pi r^2\). \(A = 3.14 \times 9 = \mathbf{28.26}\) sq. units. Find the perimeter of a square with side length 7.Concept: \(P = 4s\) (four equal sides). \(P = 4 \times 7 = \mathbf{28}\) units. Find the circumference of a circle with radius 5. (Use \(\pi \approx 3.14\))Concept: \(C = 2\pi r\). \(C = 2 \times 3.14 \times 5 = \mathbf{31.4}\) units. Find the area of a parallelogram with base 10 and height 6.Concept: \(A = b \times h\) (same as rectangle). \(A = 10 \times 6 = \mathbf{60}\) sq. units. Find the perimeter of a triangle with sides 3, 4, and 5.Concept: Add all three sides. \(P = 3 + 4 + 5 = \mathbf{12}\) units. Find the volume of a rectangular prism: \(l=3,\ w=4,\ h=5\).Concept: \(V = l \times w \times h\). \(V = 3 \times 4 \times 5 = \mathbf{60}\) cubic units. A right triangle has legs 3 and 4. Find the hypotenuse.Concept: Pythagorean Theorem: \(a^2 + b^2 = c^2\). \(3^2 + 4^2 = 9 + 16 = 25 \Rightarrow c = \sqrt{25} = \mathbf{5}\). A right triangle has legs 5 and 12. Find the hypotenuse.Concept: \(a^2 + b^2 = c^2\). \(25 + 144 = 169 \Rightarrow c = \sqrt{169} = \mathbf{13}\). What is the sum of interior angles in any triangle?Concept: This is a fundamental theorem of Euclidean geometry. The angles of any triangle always add up to \(\mathbf{180°}\). Two angles are supplementary. One is 70°. What is the other?Concept: Supplementary angles sum to 180°. \(180° - 70° = \mathbf{110°}\). Two angles are complementary. One is 35°. What is the other?Concept: Complementary angles sum to 90°. \(90° - 35° = \mathbf{55°}\). Vertical angles are always ___.Concept: Vertical angles are formed by two intersecting lines, opposite each other. Vertical angles are always equal (congruent). An exterior angle of a triangle equals 100°. What is the sum of the two non-adjacent interior angles?Concept: Exterior Angle Theorem: exterior angle = sum of the two remote interior angles. The two remote interior angles sum to \(\mathbf{100°}\). A right triangle has hypotenuse 10 and one leg 6. Find the other leg.Concept: Rearrange: \(b = \sqrt{c^2 - a^2}\). \(\sqrt{10^2 - 6^2} = \sqrt{100-36} = \sqrt{64} = \mathbf{8}\). Find the mean of: 2, 4, 6, 8.Concept: Mean = sum of all values ÷ number of values. \(\dfrac{2+4+6+8}{4} = \dfrac{20}{4} = \mathbf{5}\). Find the median of: 1, 3, 5, 7, 9.Concept: Median = middle value when data is ordered. With 5 values, it's the 3rd. Ordered: 1, 3, 5, 7, 9. Median = 5. Find the mode of: 2, 3, 2, 5, 3, 2.Concept: Mode = the value that appears most often. 2 appears 3 times, 3 appears twice. Mode = 2. Find the range of: 4, 9, 2, 7, 1.Concept: Range = maximum − minimum. \(9 - 1 = \mathbf{8}\). What is the probability of rolling an even number on a fair six-sided die?Concept: \(P = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}\). Even numbers: 2, 4, 6. \(P = \dfrac{3}{6} = \mathbf{\dfrac{1}{2}}\). A bag has 3 red and 2 blue marbles. What is the probability of picking red?Concept: Total marbles = 3+2 = 5. \(P(\text{red}) = \dfrac{3}{5}\). Find the mean of: 10, 20, 30.Concept: Mean = sum ÷ count. \(\dfrac{10+20+30}{3} = \dfrac{60}{3} = \mathbf{20}\). If \(P(A) = 0.4\), what is \(P(\text{not } A)\)?Concept: Complement Rule: \(P(\text{not } A) = 1 - P(A)\). \(1 - 0.4 = \mathbf{0.6}\).

Pre-Algebra Mastery 0 / 100 ⎙ Print PDF

Pre-Algebra
Mastery Test

Work through every essential topic — from integers to probability. Instant feedback after each answer.

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