Solve linear inequalities in one variable

Warm-up

Three quick questions from earlier lessons.

Discover

$-2x > 6$ . Try $x = -4$ : $-2 \cdot (-4) = 8 > 6$ . So which describes all solutions?

$x < -3$
$x > -3$

Key idea

Solve inequalities like equations, with one extra rule.

Adding, subtracting, multiplying or dividing by a positive number keeps the inequality. Multiplying or dividing by a negative number flips it. Strict signs (

<

,

>

) exclude the endpoint;

\le

,

\ge

include it.

Worked example

Solve $-3x + 4 \le 13$ .

-3x \le 9

Subtract 4.

x \ge -3

Divide by −3: flip.

[-3, \infty)

Endpoint included.

Your turn, guided

Your turn: $2x - 1 < 9$ . The boundary value of $x$ .

x < 5

2x < 10.

Practice ★

Solve $8x + 5 < 45$ . Give the solution set as an interval.
Solve $6x + 5 < 59$ . Give the solution set as an interval.

You have 109 dollars and tickets cost 9 dollars each. Solve $9t \le 109$ : what is the greatest number of tickets you can buy?

Watch out!

Practice ★★

Solve $-2x + 17 \le 9$ . Give the solution set as an interval.
Solve $-2x + 1 \le 19$ . Give the solution set as an interval.

Exit ticket

Three questions to check you have got it.

Solve $8x + 5 < 45$ . Give the solution set as an interval.
Solve $6x + 5 < 59$ . Give the solution set as an interval.
Solve $7x + 9 < -54$ . Give the solution set as an interval.

Summary card

Inequalities

Solve as an equation; flip when multiplying or dividing by a negative.

Watch out: check with a number from your answer.

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