SAT Math: Linear inequalities

Linear inequalities in one or two variables.

Algebra · Linear inequalities in one or two variables
Both modules
13 practice questions · 4 with free solutions
Calculator allowed (Desmos)

Practise linear inequalities →Adaptive SAT module

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 $\leq$ ; 'at least' is $\geq$ .
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

Which integer is the smallest possible value of $x$ if $6 - 3 x < - 7$ ?

Student-produced response: type your answer.

Answer: 5

$- 3 x < - 13$ , so $x > \frac{13}{3}$ (dividing by $- 3$ reverses the inequality). The smallest integer is 5.

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Solutions to the first 3 are open to everyone; the rest need only a free account. Every answer here was checked independently before it was published.

Question 1

Which integer is the smallest possible value of $x$ if $4 - 2 x < - 4$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 5

$- 2 x < - 8$ , so $x > 4$ (dividing by $- 2$ reverses the inequality). The smallest integer is 5.

Question 2

Which point $(x, y)$ satisfies both $y > 2 x - 3$ and $y < - x + 4$ ?

$(3, 1)$
$(4, - 1)$
$(0, 0)$
$(- 1, 6)$

Show the answer and solution

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.

Question 3

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: $37$

$80 + 30 n \leq 1200$ , so $n \leq \frac{1120}{30} \approx 37.33$ . The greatest whole number is 37.