SAT Math: Linear inequalities
Linear inequalities in one or two variables.
- Algebra · Linear inequalities in one or two variables
- Both modules
- 13 practice questions
- Calculator allowed (Desmos)
What the test covers
- Solution sets and which point satisfies a system
- Inequalities from word constraints
Key ideas
- Solve like an equation, but reverse the sign when multiplying or dividing by a negative.
- 'At most' is \(\le\); 'at least' is \(\ge\).
- For a point to satisfy a system, it must satisfy every inequality: test each option.
Common mistakes
- The greatest whole number of items is the floor of the bound, not the rounded value.
- Strict inequalities exclude the boundary.
Do it in Desmos
Graph the inequalities; Desmos shades the region, and the solution set is the overlap.
Worked example
Worked example
Which point \((x,y)\) satisfies both \(y>2x-3\) and \(y<-x+4\)?
- \((3, 1)\)
- \((4, -1)\)
- \((0, 0)\)
- \((-1, 6)\)
Answer: C: \((0, 0)\)
\((0,0)\): \(0>-3\) and \(0<4\), both true. \((3,1)\): \(1>3\) false. \((-1,6)\): \(6<5\) false. \((4,-1)\): \(-1>5\) false.
Practice questions
Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.
Question 1
A van can carry at most 900 kilograms. The driver weighs 80 kilograms and each box weighs 35 kilograms. What is the greatest number of boxes the van can carry with the driver?
- \(26\)
- \(24\)
- \(25\)
- \(23\)
Show the answer and solution
Answer: D: \(23\)
\(80+35n\le900\), so \(n\le\tfrac{820}{35}\approx23.4\). The greatest whole number is 23.
Question 2
Which integer is the smallest possible value of \(x\) if \(7-3x<-11\)?
Show the answer and solution
Answer: 7
\(-3x<-18\), so \(x>6\) (dividing by \(-3\) reverses the inequality). The least integer is 7.
Question 3
A lift can carry at most 900 kilograms. The operator weighs 90 kilograms and each crate weighs 40 kilograms. What is the greatest number of crates the lift can carry with the operator?
- \(22\)
- \(21\)
- \(19\)
- \(20\)
Show the answer and solution
Answer: D: \(20\)
\(90+40n\le900\), so \(n\le\tfrac{810}{40}\approx20.25\). The greatest whole number is 20.
Question 4
A lift can carry at most 800 kilograms. The operator weighs 70 kilograms and each crate weighs 40 kilograms. What is the greatest number of crates the lift can carry with the operator?
- \(18\)
- \(17\)
- \(19\)
- \(20\)
Show the answer and solution
Answer: A: \(18\)
\(70+40n\le800\), so \(n\le\tfrac{730}{40}\approx18.25\). The greatest whole number is 18.
Question 5
A lift can carry at most 1200 kilograms. The operator weighs 80 kilograms and each crate weighs 30 kilograms. What is the greatest number of crates the lift can carry with the operator?
- \(40\)
- \(36\)
- \(37\)
- \(38\)
Show the answer and solution
Answer: C: \(37\)
\(80+30n\le1200\), so \(n\le\tfrac{1120}{30}\approx37.33\). The greatest whole number is 37.
Question 6
A lift can carry at most 1000 kilograms. The operator weighs 80 kilograms and each crate weighs 30 kilograms. What is the greatest number of crates the lift can carry with the operator?
- \(31\)
- \(30\)
- \(33\)
- \(29\)
Show the answer and solution
Answer: B: \(30\)
\(80+30n\le1000\), so \(n\le\tfrac{920}{30}\approx30.67\). The greatest whole number is 30.
Question 7
A lift can carry at most 1000 kilograms. The operator weighs 90 kilograms and each crate weighs 45 kilograms. What is the greatest number of crates the lift can carry with the operator?
- \(21\)
- \(20\)
- \(19\)
- \(22\)
Show the answer and solution
Answer: B: \(20\)
\(90+45n\le1000\), so \(n\le\tfrac{910}{45}\approx20.22\). The greatest whole number is 20.
Question 8
Which integer is the smallest possible value of \(x\) if \(6-3x<-7\)?
Show the answer and solution
Answer: 5
\(-3x<-13\), so \(x>\frac{13}{3}\) (dividing by \(-3\) reverses the inequality). The smallest integer is 5.
Question 9
Which integer is the smallest possible value of \(x\) if \(2-4x<-15\)?
Show the answer and solution
Answer: 5
\(-4x<-17\), so \(x>\frac{17}{4}\) (dividing by \(-4\) reverses the inequality). The smallest integer is 5.
Question 10
Which integer is the smallest possible value of \(x\) if \(6-2x<-16\)?
Show the answer and solution
Answer: 12
\(-2x<-22\), so \(x>11\) (dividing by \(-2\) reverses the inequality). The smallest integer is 12.
Question 11
Which integer is the smallest possible value of \(x\) if \(5-3x<1\)?
Show the answer and solution
Answer: 2
\(-3x<-4\), so \(x>\frac{4}{3}\) (dividing by \(-3\) reverses the inequality). The smallest integer is 2.
Question 12
Which integer is the smallest possible value of \(x\) if \(4-2x<-4\)?
Show the answer and solution
Answer: 5
\(-2x<-8\), so \(x>4\) (dividing by \(-2\) reverses the inequality). The smallest integer is 5.
Keep going
- Previous topic: Systems of linear equations
- Next topic: Equivalent expressions
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