Before you read

Answer first — see what you already know.

Which point is in the solution set of the system?
$⎩ ⎨ ⎧ y \geq 0 x + y \leq 5 x \geq 0$

A
$(6, 0)$
B
$(2, 4)$
C
$(2, 2)$
D
$(- 1, 3)$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

${y > x y < - x + 6$
Which point is in the solution set?

A
$(1, 1)$
B
$(2, 3)$
C
$(3, 4)$
D
$(4, 1)$

Check your answer

Answer:

Theory

Systems of Inequalities

A system of inequalities is two or more inequalities that must be satisfied simultaneously.

The solution is the overlap region — the area on the graph where ALL shaded regions intersect.

System of inequalities: solution is the overlap region

$\begin{cases} y \geq x - 1 \\ y \leq -x + 5 \end{cases}$

Each inequality defines a half-plane. The solution set is where both half-planes overlap.

Theory

How to Solve Graphically

1. Graph each inequality separately (with correct shading and dashed/solid lines)
2. Find the overlap region — this is the solution
3. Any point in the overlap satisfies all inequalities

With Desmos: type both inequalities. Desmos shows the overlap as the darkest shaded region. This is by far the fastest method on the SAT.

Example 1

Find a point in the solution set of:
$\begin{cases} y > -2 \\ x + y < 6 \end{cases}$

$y > -2$ : everything above the line $y = -2$ .

$x + y < 6$ : everything below the line $x + y = 6$ .

Overlap: the region above $y = -2$ AND below $x + y = 6$ .

Example point: $(0, 0)$ . Check: $0 > -2$ ✓ and $0 + 0 = 0 < 6$ ✓.

(0, 0)

is one of many solutions.

Continue this lesson in the app

3 more sections including examples, practice problems, and step-by-step solutions.

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