Linear inequalities and systems of inequalities in two variables

Warm-up

Three quick questions from earlier lessons.

Discover

Is the point $(0, 0)$ a solution of $y > 2x + 3$ ?

No
Yes

Key idea

An inequality in two variables is a half-plane.

Graph

y < mx + b

or

y > mx + b

with a dashed boundary line, and

\le

or

\ge

with a solid line. Shade above for

y >

and below for

y <

(or test a point). The solutions of a system of inequalities are where the shadings overlap.

Worked example

$y \ge -x + 4$ .

Solid line

y = -x + 4

≥ includes the line.

Shade the side without the origin: above

Test (0, 0): 0 ≥ 4 is false.

Your turn, guided

Your turn: is $(1, 6)$ a solution of $y > 3x + 2$ ? Type 1 for yes, 0 for no.

1

6 > 5: yes.

Practice ★

Which point is a solution of $y > 3x + 3$ ?
1. $(0, 4)$
2. $(0, 3)$
3. $(0, 2)$
Which point is a solution of $y > 5x + 7$ ?
1. $(0, 8)$
2. $(0, 7)$
3. $(0, 6)$

How do you graph $y \le 6x + 2$ ?
1. Solid line, shade below
2. Dashed line, shade below
3. Solid line, shade above

Watch out!

Practice ★★

Adult tickets cost 15 dollars and child tickets 5 dollars. A group can spend at most 168 dollars on $x$ adult and $y$ child tickets. Which inequality models this?
1. $15x + 5y \le 168$
2. $15x + 5y \ge 168$
3. $15x + 5y < 168$
Adult tickets cost 9 dollars and child tickets 3 dollars. A group can spend at most 126 dollars on $x$ adult and $y$ child tickets. Which inequality models this?
1. $9x + 3y \le 126$
2. $9x + 3y \ge 126$
3. $9x + 3y < 126$

Exit ticket

Three questions to check you have got it.

Which point is a solution of $y > 3x + 3$ ?
1. $(0, 4)$
2. $(0, 3)$
3. $(0, 2)$
Which point is a solution of $y > 5x + 7$ ?
1. $(0, 8)$
2. $(0, 7)$
3. $(0, 6)$
Which point is a solution of $y > 3x + 8$ ?
1. $(0, 9)$
2. $(0, 8)$
3. $(0, 7)$

Summary card

Inequalities in two variables

Dashed for $<, >$ ; solid for $\le, \ge$ . Test a point to choose the side.

Watch out: “at most” is $\le$ , “at least” is $\ge$ .

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