Question 1
What is the least integer value of x that satisfies 2x - 3 > -20?
- A.
-9
- B.
-8
- C.
-7
- D.
8
Answer: B. -8
Add 3: 2x > -17. Divide by 2: x > -17/2. The least integer greater than -17/2 is -8.
SAT · Math · Algebra
Linear inequalities in one or two variables
Linear inequalities in one or two variables is part of Algebra in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include solve a one-variable inequality, budget or capacity constraint, point satisfying a system of inequalities and graph of an inequality.
Common traps: not flipping the sign when dividing by a negative; boundary included or excluded wrongly; dropping a negative when combining terms.
Sample questions
Question 2
A film crew's daily cost, in dollars, is 250 for equipment hire plus 20 for catering per person. With p people on set, the producer needs the total cost to be at least $650 and at most $1,050. Which compound inequality gives all possible values of p?
- A.
20 ≤ p ≤ 40
- B.
32.5 ≤ p ≤ 52.5
- C.
400 ≤ p ≤ 800
- D.
45 ≤ p ≤ 65
Answer: A. 20 ≤ p ≤ 40
The cost is 250 + 20p, so 650 ≤ 250 + 20p ≤ 1,050. Subtracting 250 gives 400 ≤ 20p ≤ 800, and dividing by 20 gives 20 ≤ p ≤ 40 (A). B skips subtracting 250. C subtracts but forgets to divide by 20. D adds 250 instead of subtracting it.
- A.
Question 3
A bakery uses 12 kilograms of flour for each batch of bread, b, and 3 kilograms for each batch of biscuits, c. It has at most 180 kilograms of flour a week, so 12b + 3c ≤ 180. If the bakery makes 12 batches of bread in a week, which statement must be true?
- A.
It can make at most 12 batches of biscuits.
- B.
It must make at least 12 batches of biscuits.
- C.
It can make at most 36 batches of biscuits.
- D.
It can make at most 60 batches of biscuits.
Answer: A. It can make at most 12 batches of biscuits.
With b = 12, the bread uses 144 kilograms, so 3c ≤ 36 and c ≤ 12: at most 12 batches of biscuits (A). B reverses the inequality; C stops at the 36 kilograms of flour left instead of dividing by 3; D ignores the bread and divides all 180 kilograms by 3.
- A.
Question 4
If 7 − 2k > 15, which of the following must be true?
- A.
3k + 1 < −13
- B.
3k + 1 > −11
- C.
3k + 1 > 13
- D.
3k + 1 < −11
Answer: D. 3k + 1 < −11
Subtracting 7 gives −2k > 8, and dividing by −2 flips the sign: k < −4. Then 3k < −12 and 3k + 1 < −11 (D). A subtracts 1 instead of adding it; B forgets to flip the sign when dividing by −2; C drops the negative entirely (k > 4, so 3k + 1 > 13).
- A.
Question 5
In the xy-plane, a shaded region lies above a dashed line through (0, 3) and (3, 0) and on or below a solid line through (0, −1) and (2, 3). Which system of inequalities does the shaded region represent?
- A.
y ≥ −x + 3 and y < 2x − 1
- B.
y > x + 3 and y ≤ 2x − 1
- C.
y > −x + 3 and y ≤ 2x − 1
- D.
y < −x + 3 and y ≥ 2x − 1
Answer: C. y > −x + 3 and y ≤ 2x − 1
The dashed line has slope −1 and intercept 3, and "above, dashed" is y > −x + 3. The solid line has slope 4/2 = 2 and intercept −1, and "on or below" is y ≤ 2x − 1, giving C. A swaps which boundary is strict; B gives the first line slope +1; D shades the wrong side of both lines.
- A.
Find your real gaps
A few questions show the style. The free scan shows which SAT skills cost you the most points, then builds a plan around them.