SAT Math skill page

SAT Systems of Inequalities Practice

Use multiple inequalities together to decide which values or choices satisfy every SAT constraint.

Practice systems of inequalities in Dolphin SAT Practice questions

12-18 min practice time3 questions on pageHeart of Algebra

Practice time12-18 min

Question bank3 questions

Best forHeart of Algebra

What this tests

What to know for this SAT skill

Check whether a value satisfies multiple inequalities
Translate paired constraints from word problems
Find overlap between solution ranges
Use boundary lines and shaded regions on graphs

Dolphin question bank

Practice your skill

Question 1 of 3Hard

Which of the following consists of the y-coordinates of all the points that satisfy the system of inequalities above? y > 2x - 1 2x > 5

y > 6
y > 4
$y > \frac{5}{2}$
$y > \frac{3}{2}$

Show answer and solution

Correct answerB. y > 4

Solution walkthrough

Subtracting the same number from each side of an inequality gives an equivalent inequality.
Hence, subtracting 1 from each side of the inequality $2 x > 5$ gives $2 x - 1 > 4$ .
So the given system of inequalities is equivalent to the system of inequalities $y > 2 x - 1$ and $2 x - 1 > 4$ , which can be rewritten as $y > 2 x - 1 > 4$ .
Using the transitive property of inequalities, it follows that $y > 4$ .

Why the other choices miss

Choice A is incorrect because there are points with a

y

-coordinate less than 6 that satisfy the given system of inequalities. For example,

(3, 5.5)

satisfies both inequalities. Choice C is incorrect. This may result from solving the inequality

2 x > 5

for

x

, then replacing

x

with

y

. Choice D is incorrect because this inequality allows

y

-values that are not the

y

-coordinate of any point that satisfies both inequalities. For example,

y = 2

is contained in the set

y > \frac{3}{2}

; however, if 2 is substituted into the first inequality for

y

, the result is

x \frac{5}{2}

Question 2 of 3Medium

Which of the following points satisfies the system of

in e q u a l i t i es 3 x + y \leq 4

and

x - y \geq 2

(2, 1)
(2, -3)
(1, 5)
(3, 1)

Show answer and solution

Correct answerB. (2, -3)

Solution walkthrough

Test each point in the system of inequalities.
Choice A, $(2, 1)$ , fails the first inequality because $3 (2) + 1 = 7$ , and 7 is not less than or equal to 4.
For choice B, $(2, - 3)$ , the first inequality gives $3 (2) + (- 3) = 3 \leq 4$ , and the second inequality gives $2 - (- 3) = 5 \geq 2$ .
Since choice B satisfies both inequalities, it is the correct answer.
Choices C and D do not need to be checked once a valid point has been found.

Question 3 of 3Medium

Which of the following ordered pairs (x, y) satisfies the system of inequalities below?

y \leq 3 x + 1

x - y > 1

(-2, -1)
(-1, 3)
(1, 5)
(2, -1)

Show answer and solution

Correct answerD. (2, -1)

Solution walkthrough

Any point $(x, y)$ that is a solution to the given system of inequalities must satisfy both inequalities in the system. The second inequality in the system can be rewritten as $x > y + 1$ .
Of the given answer choices, only choice D satisfies this inequality, because inequality $2 > - 1 + 1$ is a true statement. The point $(2, - 1)$ also satisfies the first inequality.
Alternate approach: Substituting $(2, - 1)$ into the first inequality gives $- 1 \leq 3 (2) + 1$ , or $- 1 \leq 7$ , which is a true statement. Substituting $(2, - 1)$ into the second inequality gives $2 - (- 1) > 1$ , or $3 > 1$ , which is a true statement.
Therefore, since $(2, - 1)$ satisfies both inequalities, it is a solution to the system.

Why the other choices miss

Choice A is incorrect because substituting

- 2

for

x

and

- 1

for

y

in the first inequality gives

- 1 \leq 3 (- 2) + 1

, or

- 1 \leq - 5

, which is false. Choice B is incorrect because substituting

- 1

for

x

and

3

for

y

in the first inequality gives

3 \leq 3 (- 1) + 1

, or

3 \leq - 2

, which is false. Choice C is incorrect because substituting

1

for

x

and

5

for

y

in the first inequality gives

5 \leq 3 (1) + 1

, or

5 \leq 4

, which is false.

Avoid these traps

Common mistakes on this skill

Accepting a point that satisfies only one inequality

A solution to a system must lie in the overlap and satisfy every inequality simultaneously.

Using the wrong boundary style

Use a solid boundary for <= or >= and a dashed boundary for < or >.

Study plan

How to practice this skill in Dolphin

Graph each boundary line and decide whether that boundary is included.
Use a test point to determine the correct side of each boundary.
Identify the overlap and substitute the requested point into every original inequality.

Practice systems of inequalities in Dolphin SAT

Related practice

Build the surrounding skills

Skill cluster

Keep practicing SAT Math

FAQ

Questions about SAT Systems of Inequalities Practice

What is the solution to a system of inequalities?

It is the set of points that satisfies all inequalities in the system at the same time.

How do I choose which side of a line to shade?

Substitute a test point not on the line. Shade the side containing it only if the inequality is true.