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 stained-glass maker's budget for a window is 330 dollars. A frame costs 60 dollars and each triangular pane costs 18 dollars, so the number of panes, n, satisfies 18n + 60 ≤ 330. Which statement best interprets the solution n ≤ 15?
- A.
The window must have exactly 15 triangular panes to use the budget.
- B.
The window must have at least 15 triangular panes to meet the budget.
- C.
The window can have at most 15 triangular panes without going over budget.
- D.
Each triangular pane must cost at most 15 dollars.
Answer: C. The window can have at most 15 triangular panes without going over budget.
Subtract 60 and divide by 18: n ≤ 270/18 = 15. So 15 is the greatest number of panes the budget allows, and fewer panes also work. A treats the inequality as an equation; B reverses the direction of the inequality; D confuses n, the number of panes, with the price of a pane.
- A.
Question 3
Which inequality is equivalent to −5x + 4 > 29?
- A.
x < −5
- B.
x > −5
- C.
x > 5
- D.
x < 5
Answer: A. x < −5
Subtract 4 to get −5x > 25, then divide by −5 and reverse the inequality: x < −5 (choice A). Choice B forgets to reverse the sign; C and D lose the negative sign, treating the quotient as +5.
- A.
Question 4
What is the greatest integer x that satisfies −4(x − 3) + 2 > 26 − x?
Enter your own answer.
Answer: -5
Distributing gives −4x + 14 > 26 − x. Adding x and subtracting 14 gives −3x > 12, and dividing by −3 flips the sign: x < −4. The greatest integer less than −4 is −5. Answering −4 includes the excluded boundary, and forgetting to flip the sign gives x > −4, which has no greatest integer.
Question 5
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.
Find your real gaps
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