Topic-specific practice

Work. Multiply by 4 to obtain 3x −5 = 28. The requested expression is 7 greater than this known expression: 3x + 2 = (3x − 5) + 7 = 35. Finding x = 11 is valid but unnecessary.

Answer: 35

Check. Substitution gives 3(11) + 2 = 35 and (33 − 5)/4 = 7.

Key takeaway. Topic practice should include a change of target, not just new coefficients. Solve for a useful group when the problem gives one.

Work. The slope is (3 − 11)/(6 − 2) = −8/4 = −2. In y = −2x + b, use (2, 11): 11 = −4 + b, so b = 15. The intercept is the output when x = 0, not the first output listed.

Answer: 15

Check. At x = 6, the rule gives −12 + 15 = 3.

Key takeaway. Practice the same relationship as a table, an equation, and a graph. A table row is not automatically an intercept.

Work. The left side of the second equation is 3 times the first. Its right side must also be 3 times the first: k = 3(9) = 27. Otherwise the lines are distinct and parallel.

Answer: 27

Check. With k = 27, division of the second equation by 3 reproduces the first equation exactly.

Key takeaway. The three proportional quantities are both variable coefficients and the constant. Matching slopes alone does not establish the same line.

Work. Let n be a nonnegative integer. The budget is 25 + 12n ≤ 157, so 12n ≤ 132 and n ≤ 11. The greatest admissible integer is 11.

Answer: 11 participants

Check. Eleven cost $157; twelve cost $169, exceeding the budget.

Key takeaway. For an integer maximum, test the proposed integer and the next one. Do not round an over-budget answer to the nearest integer.

Work. Use x² − 16 = (x − 4)(x + 4). Cancel the nonzero factor x − 4 to obtain x + 4, retaining x ≠ 4. At x = 7, the value is 11.

Answer: x + 4 for x ≠ 4; value 11

Check. The original expression at 7 is (49 − 16)/(7 − 4) = 33/3 = 11. At 4 it remains undefined.

Key takeaway. An equivalent formula and an equivalent domain belong together. Cancellation does not restore an excluded input.

Work. The denominator 3 means a cube root; the numerator 2 means a square. Thus 64^2/3 = ($\sqrt[\text{3}]{\text{64}}$)² = 4² = 16. Because the base is positive, these real-power manipulations are valid.

Answer: 16

Check. 16^3/2 = ($\sqrt{\text{16}}$)³ = 4³ = 64.

Key takeaway. Explain the role of numerator and denominator rather than memorizing a keystroke.

Work. A square is nonnegative, so 2(x − 4)² ≥ 0. The smallest possible output is −7, reached when x = 4. The input 4 and the output −7 answer different questions.

Answer: −7

Check. f(4) = −7 and f(3) = f(5) = −5, both higher.

Key takeaway. Practice distinguishing the vertex input, vertex output, and intercepts. The sign of the leading coefficient determines minimum versus maximum.

Work. The right side must be nonnegative, so x ≥ −1. Squaring gives x + 13 = x² + 2x + 1, or x² + x − 12 = (x + 4)(x − 3) = 0. The candidates are −4 and 3; only 3 satisfies the restriction.

Answer: x = 3

Check. At 3, both sides equal 4. At −4, the left side equals 3 but the right side equals −3.

Key takeaway. The goal is not merely to solve the transformed equation. Every candidate must satisfy the original relationship.

Work. Increasing t by 3 increases the exponent by 1, so the value is multiplied by 1.12. This is a 12% increase over each 3-month interval. The equivalent one-month multiplier would be 1.12^1/3.

Answer: 12%

Check. N(3)/N(0) = 1.12. The coefficient 240 cancels in the ratio.

Key takeaway. An exponential base belongs to an interval. Do not call 12% a monthly growth rate when the exponent is t/3.

Work. The two totals are 8(70) = 560 and 12(85) = 1020. Combine them and divide by the combined count: (560 + 1020)/20 = 79.

Answer: 79

Check. The result lies between 70 and 85 and is closer to 85 because that group is larger.

Key takeaway. Weighted means are ordinary means after totals and counts have been reconstructed.

Work. The selection is restricted to the 24 music-club members. Nine satisfy the requested condition, so the probability is 9/24 = 3/8. The total enrollment 60 is not the sample space for this selection.

Answer: $\frac{\text{3}}{\text{8}}$

Check. The numerator 9 is part of the denominator group 24, so the fraction is between 0 and 1.

Key takeaway. Write the conditioning phrase beside the denominator before performing any arithmetic.

Work. The mean increases by 6. Each new deviation from the new mean is (x_i + 6) − ($\overline{x}$ + 6) = x_i − $\overline{x}$, identical to its old deviation. Therefore the standard deviation is unchanged.

Answer: Mean increases by 6; standard deviation is unchanged.

Check. The distance between every pair of observations is unchanged by the translation.

Key takeaway. Do not infer a greater spread from larger individual values. Spread describes separation, not position.

Work. No. Random sampling supports an inference about the sampled town, subject to uncertainty and study quality. It does not assign commuting conditions. Other variables may explain some or all of the association.

Answer: An association can be studied; causation is not established by this survey.

Check. Income, work hours, or household obligations could affect both quantities without being controlled.

Key takeaway. Random sampling addresses whom a result may represent. Random assignment addresses a causal comparison; these are different design decisions.

Work. The area ratio is 125/45 = 25/9. The larger-to-smaller length factor is $\sqrt{\text{25/9}}$ = 5/3. Therefore s = 9(5/3) = 15.

Answer: 15

Check. The squared side factor is (15/9)² = 25/9, the area factor.

Key takeaway. Area, length, and volume do not use the same scale factor. Take the appropriate root before scaling a length.

Work. Complete both squares: (x − 4)² − 16 + (y + 3)² − 9 = 11. Hence (x − 4)² + (y + 3)² = 36. The center is (4, −3) and the radius is $\sqrt{\text{36}}$ = 6.

Answer: 6

Check. At (10, −3), a point 6 units from the center, the original left side equals 11.

Key takeaway. The constant on the completed-square side is the radius squared. Both completed squares change the constant.