Inequality questions on Digital SAT Math test the same algebra as equations, plus a few extra layers: translating phrases like "at least" and "no more than" into the correct inequality symbol, reading a shaded region on a graph to find valid (x, y) solutions, working with systems of two inequalities at once, and remembering to flip the inequality sign when multiplying or dividing by a negative number. The underlying algebra is rarely difficult — the challenge is almost always translation and sign-tracking.
The 23 problems below cover the full range of how inequalities appear on real, recently administered Digital SAT exams, including several shaded-region graph problems, systems of two inequalities, and real-world constraint word problems (weight limits, budget minimums, temperature ranges). Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. For the graph-based problems, the fastest approach is usually to test each answer choice directly against the boundary line or shaded region rather than trying to derive the full inequality from scratch. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
1Counting Solutions to a System
y = 5x and y = 2x + 2. How many solutions does the given system of equations have?
A) Exactly one
B) Exactly two
C) Infinitely many
D) Zero
Answer Explanation
Set the two expressions for y equal: 5x = 2x + 2, so 3x = 2, giving x = 2/3. This is one specific value of x, so there is one specific point where the lines meet.
Since the two lines have different slopes (5 and 2), they are not parallel and must cross at exactly one point. The correct answer is A.
2Identifying a Region with No Solutions
In the xy-plane, which of the following does NOT contain any points (x, y) that are solutions to 7x + 4y > 12?
A) The region where x > 0 and y > 0
B) The region where x < 0 and y > 0
C) The region where x < 0 and y < 0
D) The region where x > 0 and y < 0
Answer Explanation
In quadrants where x > 0 or y > 0, you can always pick a large enough positive value to push 7x + 4y above 12 — so those regions do contain solutions.
But when both x < 0 and y < 0, both terms 7x and 4y are negative, so their sum 7x + 4y is always negative — which can never be greater than 12. The correct answer is C.
3Writing an Inequality from a Word Problem
In December 2017, the lowest temperature recorded in a certain city was 40°F and the highest temperature recorded was 90°F. Which inequality is true for all values of t, where t represents any temperature in °F recorded in the city in December?
A) 40 ≤ t ≤ 90
B) t ≤ 40
C) t ≤ 50
D) t ≥ 90
Answer Explanation
The temperature never went below the recorded minimum (40°F) or above the recorded maximum (90°F), so every recorded value falls between the two, inclusive.
That gives 40 ≤ t ≤ 90. The correct answer is A.
4Testing Table Values Against an Inequality
y ≤ (2/3)x + 5. For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?
A)
x
y
-1.5
6
1.5
5
3
-4
B)
x
y
-1.5
5
1.5
7
3
8
C)
x
y
-1.5
6
1.5
5
3
6
D)
x
y
-1.5
3
1.5
5
3
6
Answer Explanation
Check each x-value against the boundary: at x = −1.5, (2/3)(−1.5)+5 = 4, so y must be ≤ 4. At x = 1.5, the boundary is 6, so y ≤ 6. At x = 3, the boundary is 7, so y ≤ 7.
Only one table has all three y-values at or below these three boundary values (3 ≤ 4, 5 ≤ 6, and 6 ≤ 7) — every other table has at least one y-value that exceeds its boundary. The correct answer is D.
5Writing an Inequality from a Word Problem
A model estimates that a gray whale travels 73 to 78 miles each day during its migration. Based on this model, which inequality represents the estimated total number of miles, x, a gray whale could travel in 31 days of its migration?
A) (73)(31) ≤ x ≤ (78)(31)
B) 73 + 31 ≤ x ≤ 78 + 31
C) 73 ≤ 31x ≤ 78
D) 73 ≤ 31 + x ≤ 78
Answer Explanation
Over 31 days, the total distance is the daily distance multiplied by 31 days, applied to both the low and high ends of the daily range.
That gives a total between 73 × 31 miles and 78 × 31 miles: (73)(31) ≤ x ≤ (78)(31). The correct answer is A.
6Reading a Solution from a Shaded Region
The shaded region shown represents solutions to an inequality. Which ordered pair (x, y) is a solution to this inequality?
Shaded region representing solutions to an inequality
A) (−7, 0)
B) (0, −7)
C) (0, 7)
D) (7, 0)
Answer Explanation
The boundary line passes through (0, 8) and (4, 0), giving the equation y = −2x + 8. The shaded region lies to the right of this steep line, which corresponds to points where y > −2x + 8.
Testing (7, 0): is 0 > −2(7) + 8 = −6? Yes, that's true. None of the other three points satisfy this inequality. The correct answer is D.
7Writing an Inequality from a Description
The minimum value of x is 11 less than 8 times another number n. Which inequality shows the possible values of x?
A) x ≤ 8n − 11
B) x ≥ 8n − 11
C) x ≤ 11 − 8n
D) x ≥ 11 − 8n
Answer Explanation
"11 less than 8 times n" translates to 8n − 11. Since this quantity is described as the minimum value x can take, x must be greater than or equal to it.
That gives x ≥ 8n − 11. The correct answer is B.
8Solving a System of Inequalities
y ≤ x + 9 and y ≥ −2x − 9. Which point (x, y) is a solution to the given system of inequalities in the xy-plane?
A) (0, −10)
B) (0, 10)
C) (−7, 0)
D) (7, 0)
Answer Explanation
Test each point against both inequalities. For (7, 0): y ≤ x + 9 gives 0 ≤ 16, true. y ≥ −2x − 9 gives 0 ≥ −23, also true.
The other three points each fail at least one of the two inequalities. The correct answer is D.
9Interpreting a Vertex in Context
The function f(x) = (1/7)(x − 5)² + 6 gives a toy car's height above the ground f(x), in inches, x seconds after it started moving on an elevated track, where 0 < x ≤ 10. Which of the following is the best interpretation of the vertex of the graph of y = f(x) in the xy-plane?
A) The toy car's minimum height was 6 inches above the ground.
B) The toy car's minimum height was 5 inches above the ground.
C) The toy car's height was 6 inches above the ground when it started moving.
D) The toy car's height was 5 inches above the ground when it started moving.
Answer Explanation
The function is in vertex form, (1/7)(x − 5)² + 6, with vertex at (5, 6). Since the coefficient 1/7 is positive, the parabola opens upward, meaning the vertex is the lowest point on the graph.
That means the car's minimum height was 6 inches, occurring 5 seconds after it started moving (not at the very start, since x = 5 is within the given domain but isn't x = 0). The correct answer is A.
10Writing an Inequality from a Word Problem
An arborist measured and recorded the heights of red maple trees in a certain area. Of the red maple tree heights recorded, the minimum was 42 feet and the maximum was 57 feet. Which inequality is true for all possible values of t, where t represents the recorded height, in feet, of a red maple tree in this area?
A) t ≤ 42
B) t ≥ 57
C) t ≥ 99
D) 42 ≤ t ≤ 57
Answer Explanation
Every recorded height falls between the minimum and the maximum, inclusive.
That gives 42 ≤ t ≤ 57. The correct answer is D.
11Finding a Constant from a Shaded Region
The shaded region shown represents the solutions to the inequality −18y < c, where c is a constant. What is the value of c?
Shaded region representing solutions to −18y < c
A) 162
B) 9
C) −9
D) −162
Answer Explanation
The dashed horizontal boundary line sits at y = −9, with the region shaded above it. Since the line is dashed (not solid), the inequality is strict, matching the strict "<" in the original.
Solve −18y = c for the boundary: −18(−9) = c, so c = 162. (Dividing the original inequality by −18 flips the sign, turning −18y < c into y > −c/18 — matching the "shaded above" region.) The correct answer is A.
12Solving a System of Inequalities
y ≤ x + 2 and y ≥ −3x − 4. Which point (x, y) is a solution to the given system of inequalities in the xy-plane?
A) (0, −8)
B) (0, 8)
C) (−8, 0)
D) (8, 0)
Answer Explanation
Test each point against both inequalities. For (8, 0): y ≤ x + 2 gives 0 ≤ 10, true. y ≥ −3x − 4 gives 0 ≥ −28, also true.
The other three points each fail at least one of the two inequalities. The correct answer is D.
13Solving a Real-World Constraint
A certain truck can tow a trailer if the combined weight of the trailer and the packages it contains is no more than 4,600 pounds. What's the maximum number of objects this truck can tow in a trailer with a weight of 600 pounds if each package weighs 120 pounds?
A) 33
B) 34
C) 38
D) 39
Answer Explanation
Let n be the number of packages. The total weight must satisfy 600 + 120n ≤ 4,600.
Subtract 600: 120n ≤ 4,000. Divide by 120: n ≤ 33.33. Since n must be a whole number of packages, the maximum is 33. The correct answer is A.
14Solving a Real-World Constraint
A certain truck can tow a trailer if the combined weight of the trailer and the packages it contains is no more than 4,200 pounds. What is the maximum number of packages the truck can tow in a trailer with a weight of 600 pounds if each package weighs 110 pounds?
A) 32
B) 33
C) 38
D) 39
Answer Explanation
Let n be the number of packages. The total weight must satisfy 600 + 110n ≤ 4,200.
Subtract 600: 110n ≤ 3,600. Divide by 110: n ≤ 32.7. Since n must be a whole number of packages, the maximum is 32. The correct answer is A.
15Solving a System of Inequalities
y ≤ x + 2 and y ≥ −4x − 6. Which point (x, y) is a solution to the given system of inequalities in the xy-plane?
A) (0, −12)
B) (0, 12)
C) (−12, 0)
D) (12, 0)
Answer Explanation
Test each point against both inequalities. For (12, 0): y ≤ x + 2 gives 0 ≤ 14, true. y ≥ −4x − 6 gives 0 ≥ −54, also true.
The other three points each fail at least one of the two inequalities. The correct answer is D.
16Writing an Inequality from a Description
In a set of four consecutive odd integers, ordered from least to greatest, the first integer is represented by x. The product of 18 and the fourth odd integer is at most 22 less than the sum of the first and third odd integers. Which inequality represents this situation?
A) 18(x + 6) ≤ x + (x + 4) − 22
B) 18(x + 6) ≥ 22 − (x + (x + 4))
C) 18(x + 4) ≤ x + (x + 3) − 22
D) 18(x + 4) ≥ 22 − (x + (x + 3))
Answer Explanation
Since consecutive odd integers differ by 2, the four integers are x, x+2, x+4, and x+6. The fourth is x+6, and the third is x+4.
"At most 22 less than" means the product is ≤ (sum − 22). The sum of the first and third is x + (x+4). So: 18(x+6) ≤ x + (x+4) − 22. The correct answer is A.
17Writing an Inequality from a Word Problem
During a portion of a flight, a small airplane's cruising speed varied between 180 miles per hour and 190 miles per hour. Which inequality best represents this situation, where s is the cruising speed, in miles per hour, during this portion of the flight?
A) s ≤ 10
B) s ≤ 180
C) s ≤ 190
D) 180 ≤ s ≤ 190
Answer Explanation
The speed stayed between the two given values for the entire portion of the flight described.
That gives 180 ≤ s ≤ 190. The correct answer is D.
18Writing an Inequality from a Word Problem
During a portion of a flight, a small airplane's cruising speed varied between 125 miles per hour and 135 miles per hour. Which inequality best represents this situation, where s is the cruising speed, in miles per hour, during this portion of the flight?
A) s ≤ 10
B) s ≤ 125
C) s ≤ 135
D) 125 ≤ s ≤ 135
Answer Explanation
The speed stayed between the two given values for the entire portion of the flight described.
That gives 125 ≤ s ≤ 135. The correct answer is D.
19Reading a Solution from a Shaded Region
The shaded region shown represents solutions to an inequality. Which ordered pair (x, y) is a solution to this inequality?
Shaded region representing solutions to an inequality
A) (−8, 0)
B) (0, 8)
C) (0, −8)
D) (8, 0)
Answer Explanation
The boundary line has a steep negative slope, passing near (0, 8) and (6, 0). The shaded region lies to the right of this line, corresponding to points where y > the line's value at that x.
Testing (8, 0) against a line like y = −(4/3)x + 8: the line's value at x = 8 is about −2.7, and 0 is indeed greater than that. The other candidate points fail this test. The correct answer is D.
20Writing an Inequality from a Word Problem
A newspaper stand sells magazines for $5 each and newspapers for $2 each. The stand needs to sell m magazines and n newspapers for a combined total value of at least $150 each day to cover operation costs. Which inequality represents this situation?
A) 5m + 2n ≤ 150
B) 5m + 2n ≥ 150
C) 7mn ≤ 150
D) 7mn ≥ 150
Answer Explanation
The total value from sales is 5 dollars per magazine times m magazines, plus 2 dollars per newspaper times n newspapers: 5m + 2n.
Since this total needs to be at least $150, the inequality uses ≥: 5m + 2n ≥ 150. The correct answer is B.
21Solving a Real-World Constraint
The water depth in a certain river is 78.7 feet. During a 30-day period, the water depth in this river is predicted to increase by a minimum of 1 ft and a maximum of 3 ft. What is the predicted maximum water depth, in feet, during this time?
A) 75.7
B) 77.7
C) 79.7
D) 81.7
Answer Explanation
The maximum predicted depth occurs when the increase is at its largest possible value: 3 feet.
78.7 + 3 = 81.7 feet. The correct answer is D.
22Testing Table Values Against a System
y < −6x − 22 and y > −4x − 7. For which of the following tables are all the values of x and their corresponding values of y solutions to the given system of inequalities?
A)
x
y
-9
27
-10
51
-12
-4
B)
x
y
-9
32
-10
36
-12
-4
C)
x
y
-9
28
-10
36
-12
48
D)
x
y
-9
30
-10
37
-12
48
Answer Explanation
Check each x-value against both boundaries. At x = −9: the first inequality requires y < 32, and the second requires y > 29 — so 29 < y < 32. At x = −10: 33 < y < 38. At x = −12: 41 < y < 50.
Only one table has all three y-values falling strictly within these three ranges (30 fits between 29 and 32, 37 fits between 33 and 38, and 48 fits between 41 and 50). The correct answer is D.
23Identifying an Inequality from a Shaded Region
The shaded region shown in the graph represents all the solutions to which inequality?
Shaded region representing all solutions to an inequality
A) x < 33
B) x ≥ 33
C) y ≤ 33
D) y ≥ 33
Answer Explanation
The boundary is a solid horizontal line at y = 33, with shading covering everything above it. A horizontal boundary means the inequality is written in terms of y alone, ruling out the x-only options.
Since the shaded region is above the line and the boundary is solid (inclusive), the inequality is y ≥ 33. The correct answer is D.
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