Linear Equations and Inequalities: Question 4

Worked solution

Part (a): solving the inequality

Start with

$17 - 5x < 2.$

Step 1: Subtract $17$ from both sides.

$-5x < 2 - 17$

$-5x < -15$

Step 2: Divide both sides by $-5$.

Dividing (or multiplying) an inequality by a negative number reverses the inequality sign:

$x > \frac{-15}{-5}$

$x > 3$

$\boxed{x > 3}$

Check (with a test value): try $x = 4$: $17 - 5(4) = 17 - 20 = -3$, and $-3 < 2$ is true. ✓ Try the boundary $x = 3$: $17 - 5(3) = 2$, and $2 < 2$ is false, so $x = 3$ is correctly excluded. ✓

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Part (b): the smallest integer $n$

Since $x > 3$ is a strict inequality, $x = 3$ is not included, so the solution set is every number greater than $3$.

The smallest integer greater than $3$ is

$\boxed{n = 4}$

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Final answers

• (a) $x > 3$
• (b) $n = 4$