22 Math: Algebraic Problem Solving Practice Questions & Answers
Every Math: Algebraic Problem Solving practice question from the GED Practice Test, with the correct answer and a short explanation.
Start practice test →1. A tile pattern grows the same way at each step: Figure 1 has 5 tiles, Figure 2 has 9 tiles, Figure 3 has 13 tiles, and Figure 4 has 17 tiles. Which expression gives the number of tiles in Figure n?
- A.5n
- B.4n + 1✓ Answer
- C.n + 4
- D.4n + 5
The count increases by a constant 4 tiles per figure, so the rule is linear with a rate of change of 4. Because Figure 1 already contains 5 tiles, the constant term must be 5 − 4 = 1, giving 4n + 1; checking, 4(1) + 1 = 5 and 4(4) + 1 = 17.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.1.c (write linear expressions as part of word-to-symbol translations)Report a problem with this question
2. Which expression is equivalent to (2x − 5)(3x + 4)?
- A.6x² − 7x − 20✓ Answer
- B.6x² + 7x − 20
- C.6x² − 7x + 20
- D.6x² − 20
Multiplying two binomials means distributing every term of the first over every term of the second (FOIL): (2x)(3x) = 6x², (2x)(4) = 8x, (−5)(3x) = −15x, and (−5)(4) = −20. Combining the like terms 8x − 15x gives −7x, so the product is 6x² − 7x − 20; the negative constant comes from a negative times a positive.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.1.d (multiply two binomials); distributive propertyReport a problem with this question
3. Which of the following is the complete factorization of 3x² − 11x − 4?
- A.(3x − 1)(x + 4)
- B.(3x + 4)(x − 1)
- C.(3x + 1)(x − 4)✓ Answer
- D.(3x − 4)(x + 1)
For a trinomial with a > 1, the two binomials must multiply back to give both the correct middle term and the correct constant. Expanding (3x + 1)(x − 4) gives 3x² − 12x + x − 4 = 3x² − 11x − 4, which matches; the other choices produce middle terms of +11x, +x, or −x.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.1.f (factor polynomial expressions, including trinomials with a > 1)Report a problem with this question
4. For which value(s) of x is the rational expression (x + 3) / (x² − 5x − 14) undefined?
- A.x = −3 only
- B.x = 7 only
- C.x = −7 and x = 2
- D.x = 7 and x = −2✓ Answer
A rational expression is undefined exactly where its denominator equals zero, because division by zero has no value. Factoring gives x² − 5x − 14 = (x − 7)(x + 2), which is zero at x = 7 and x = −2; the numerator's zero at x = −3 does not make the expression undefined.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.1.h and College Ready descriptor (rational expression undefined where the denominator equals 0)Report a problem with this question
5. The area of a trapezoid is A = ½h(b₁ + b₂). Which equation correctly solves this formula for b₁?
- A.b₁ = 2A/h − b₂✓ Answer
- B.b₁ = A/(2h) − b₂
- C.b₁ = (2A − b₂)/h
- D.b₁ = 2A − hb₂
A literal equation is solved by applying inverse operations to both sides in reverse order, exactly as with a numerical equation. Multiplying both sides by 2 gives 2A = h(b₁ + b₂), dividing by h gives 2A/h = b₁ + b₂, and subtracting b₂ isolates b₁ = 2A/h − b₂.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.2.a (solve linear equations whose coefficients are represented by letters)Report a problem with this question
6. A rental company charges $39 per day plus $0.20 for each mile driven. Nadia rented a car for 2 days and her total charge was $105.60. How many miles did she drive?
- A.69 miles
- B.276 miles
- C.138 miles✓ Answer
- D.528 miles
The situation models as 39(2) + 0.20m = 105.60, where the fixed daily charge is separate from the per-mile rate. Subtracting the fixed $78 leaves 0.20m = 27.60, and dividing by the unit rate 0.20 gives m = 138 miles.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.2.b (solve real-world problems involving linear equations)Report a problem with this question
7. A theater sold 210 tickets in one night. Adult tickets cost $12 each and student tickets cost $7 each, and the total collected was $1,995. How many adult tickets were sold?
- A.84
- B.105✓ Answer
- C.126
- D.70
Two unknowns require two equations: a + s = 210 (count) and 12a + 7s = 1,995 (money). Substituting s = 210 − a gives 12a + 1,470 − 7a = 1,995, so 5a = 525 and a = 105, which checks because 12(105) + 7(105) = 1,995.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.2.d (solve real-world problems leading to a system of two linear equations)Report a problem with this question
8. What is the solution set of the inequality −4x + 9 > 25?
- A.x > −4
- B.x < 4
- C.x > 4
- D.x < −4✓ Answer
Subtracting 9 from both sides gives −4x > 16, and dividing both sides by −4 requires reversing the inequality symbol, because multiplying or dividing an inequality by a negative number reverses the order of the two sides. The result is x < −4, which can be checked with x = −5: −4(−5) + 9 = 29 > 25.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.3.a; sign-reversal rule for multiplying or dividing an inequality by a negative numberReport a problem with this question
9. A student has $60 to spend at the campus store. She must pay a $12 lab fee, and each notebook costs $4.50. What is the greatest number of notebooks she can buy?
- A.10✓ Answer
- B.11
- C.13
- D.16
"Has $60 to spend" is an at-most constraint, so the model is 4.50n + 12 ≤ 60, giving 4.50n ≤ 48 and n ≤ 10.67. Because notebooks come only in whole units and the total may not exceed the budget, the answer must be rounded down to 10, not up.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, targets A.3.c and A.3.d (write and solve real-world linear inequalities; "at most" translates to ≤)Report a problem with this question
10. Which number-line graph shows the solution to 3x − 7 ≥ 2?
- A.An open circle at 3 with shading to the right
- B.A closed circle at 3 with shading to the left
- C.A closed circle at 3 with shading to the right✓ Answer
- D.A closed circle at −3 with shading to the right
Adding 7 to both sides gives 3x ≥ 9, and dividing by the positive number 3 keeps the direction, so x ≥ 3. A closed (filled) circle marks that 3 itself is included because the symbol is ≥ rather than >, and the shading goes right because the solutions are all values greater than 3.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.3.b (identify or graph the solution to a one-variable linear inequality on a number line)Report a problem with this question
11. A rectangular garden is 3 meters longer than it is wide, and its area is 54 square meters. What is the width of the garden?
- A.9 m
- B.6 m✓ Answer
- C.27 m
- D.12 m
Letting w be the width, the length is w + 3, so w(w + 3) = 54 becomes w² + 3w − 54 = 0, which factors as (w + 9)(w − 6) = 0. The two roots are w = −9 and w = 6, but a length cannot be negative, so only w = 6 m is valid in context (and 6 × 9 = 54).
Source: GED Assessment Guide for Educators, Mathematical Reasoning, targets A.4.a and A.4.b (write and solve quadratic equations in context; reject roots invalid in context)Report a problem with this question
12. An object is launched from a platform, and its height in feet after t seconds is h = −16t² + 48t + 64. After how many seconds does the object hit the ground?
- A.1 second
- B.1.5 seconds
- C.8 seconds
- D.4 seconds✓ Answer
Hitting the ground means the height is zero, so set −16t² + 48t + 64 = 0; dividing every term by −16 gives t² − 3t − 4 = 0, which factors as (t − 4)(t + 1) = 0. The roots are t = 4 and t = −1, and negative time is impossible, so the object lands at t = 4 seconds.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.4.a (solve one-variable quadratic equations with real solutions by an appropriate method)Report a problem with this question
13. A linear function is given by the table below. x: 2, 5, 8, 11 y: 17, 11, 5, −1 What is the slope of the line?
- A.−2✓ Answer
- B.2
- C.−1/2
- D.−6
Slope from a table is the ratio of the change in y to the change in x between any two rows: (11 − 17)/(5 − 2) = −6/3 = −2. The same ratio holds for every pair of rows, which is what confirms the relationship is linear, and the negative sign shows y decreases as x increases.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.5.b (determine the slope of a line from a graph, equation, or table)Report a problem with this question
14. A tank is draining. The volume of water in gallons after t minutes is V = 450 − 12t. What does the number −12 represent in this situation?
- A.The tank starts with 12 gallons of water.
- B.It takes 12 minutes for the tank to empty.
- C.The tank loses 12 gallons of water each minute.✓ Answer
- D.The tank gains 12 gallons of water each minute.
In the form y = mx + b, the coefficient of the variable is the slope, which is the unit rate of change of the output per one unit of input. Here −12 is gallons per minute, and the negative sign means the volume is decreasing, so the tank loses 12 gallons each minute; the 450 is the starting volume.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, targets A.5.c and A.5.e (interpret unit rate as slope and key features of a model in terms of the quantities)Report a problem with this question
15. What are the coordinates of the x-intercept of the line 3x − 4y = 24?
- A.(0, 8)
- B.(8, 0)✓ Answer
- C.(0, −6)
- D.(−6, 0)
Every point on the x-axis has a y-coordinate of 0, so the x-intercept is found by substituting y = 0: 3x − 4(0) = 24 gives 3x = 24 and x = 8, so the point is (8, 0). Substituting x = 0 instead would give the y-intercept (0, −6).
Source: GED Assessment Guide for Educators, Mathematical Reasoning, targets A.5.d and A.5.e (graph two-variable linear equations; interpret intercepts)Report a problem with this question
16. Which equation represents the line that passes through the points (−2, 7) and (4, −5)?
- A.y = 2x + 11
- B.y = −½x + 6
- C.y = −2x − 1
- D.y = −2x + 3✓ Answer
The slope formula gives m = (−5 − 7)/(4 − (−2)) = −12/6 = −2, and substituting one point into y = mx + b gives 7 = −2(−2) + b, so b = 3. The line y = −2x + 3 also satisfies the second point, since −2(4) + 3 = −5.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.6.b (write the equation of a line passing through two given distinct points)Report a problem with this question
17. Which equation represents the line that passes through the point (6, −1) and is perpendicular to the line y = (3/4)x + 5?
- A.y = −(4/3)x + 7✓ Answer
- B.y = (3/4)x − 5.5
- C.y = (4/3)x − 9
- D.y = −(3/4)x + 3.5
Perpendicular lines have slopes that are negative reciprocals, meaning their product is −1, so a line perpendicular to a slope of 3/4 must have slope −4/3. Using point-slope form, y + 1 = −(4/3)(x − 6) simplifies to y = −(4/3)x + 7, which contains (6, −1) because −(4/3)(6) + 7 = −1.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.6.c (use slope to identify perpendicular lines); negative-reciprocal rule m₁·m₂ = −1Report a problem with this question
18. Which equation represents the line that passes through the point (0, −3) and is parallel to the line 4x − 2y = 10?
- A.y = −2x − 3
- B.y = −½x − 3
- C.y = 2x − 3✓ Answer
- D.y = 4x − 3
Rewriting 4x − 2y = 10 in slope-intercept form gives y = 2x − 5, so its slope is 2, and parallel lines have equal slopes. Since the new line crosses the y-axis at (0, −3), its y-intercept is −3, giving y = 2x − 3.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.6.c (use slope to identify parallel lines); parallel lines have equal slopesReport a problem with this question
19. If f(x) = 2x² − 5x + 1, what is the value of f(−3)?
- A.−2
- B.34✓ Answer
- C.4
- D.16
Function notation f(−3) means substitute −3 for every x in the rule, not multiply f by −3. Since (−3)² = 9 (a negative squared is positive) and −5(−3) = +15, the value is 2(9) + 15 + 1 = 34.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.7.c (evaluate linear and quadratic functions represented in function notation)Report a problem with this question
20. Linear function f is given by the table: x = 0, 2, 4 with f(x) = 5, 11, 17. Linear function g is described this way: its graph passes through (0, 8) and (5, 18). Which statement comparing f and g is true?
- A.g has both the greater rate of change and the greater value at x = 0.
- B.f has both the greater rate of change and the greater value at x = 0.
- C.The two functions have the same rate of change.
- D.f has the greater rate of change, but g has the greater value at x = 0.✓ Answer
Rate of change is slope, computed the same way from a table or from two points: f has (11 − 5)/(2 − 0) = 3 and g has (18 − 8)/(5 − 0) = 2, so f grows faster. At x = 0 the values are f(0) = 5 and g(0) = 8, so g starts higher even though it grows more slowly.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.7.d (compare properties of two functions each represented in a different way)Report a problem with this question
21. Car P travels at a constant speed described by the equation d = 58t, where d is distance in miles and t is time in hours. Car Q also travels at a constant speed: it covers 112 miles in 2 hours and 224 miles in 4 hours. Which car is faster, and by how much?
- A.Car P, by 2 miles per hour✓ Answer
- B.Car Q, by 2 miles per hour
- C.Car P, by 54 miles per hour
- D.Neither — they travel at the same speed
To compare proportional relationships shown in different forms, convert each to its constant of proportionality (the unit rate). Car P's equation has a coefficient of 58, so it travels 58 miles per hour, while Car Q's table gives 112/2 = 56 miles per hour, making Car P faster by 2 miles per hour.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.7.a (compare two proportional relationships represented in different ways)Report a problem with this question
22. Which of the following sets of ordered pairs represents a function?
- A.{(1, 2), (1, 4), (2, 6), (3, 8)}
- B.{(4, 1), (4, −1), (5, 2), (6, 3)}
- C.{(−2, 5), (0, 5), (3, 5), (6, 5)}✓ Answer
- D.{(0, 0), (2, 3), (2, −3), (5, 7)}
A relation is a function only if each input is paired with exactly one output; repeating an output value is allowed, but repeating an input with different outputs is not. Only {(−2, 5), (0, 5), (3, 5), (6, 5)} has four different x-values, while the other sets repeat the input 1, 4, or 2 with two different y-values.
Source: GED Assessment Guide for Educators, Mathematical Reasoning, target A.7.b (identify a function in a table or graph as having exactly one output for each input)Report a problem with this question
Practice questions based on the GED Testing Service Assessment Guide for Educators and standard high-school-equivalency curriculum. GED is a registered trademark of the American Council on Education; this site is not affiliated with or endorsed by ACE or GED Testing Service. Passing scores, fees, retake limits, and age and residency eligibility are set by GED Testing Service and by individual states and change over time — confirm current requirements for your state before testing. Note that the Reasoning Through Language Arts test also includes an extended-response essay, which multiple-choice practice cannot cover. About the GED test →