Inequalities

Flipping the Inequality Sign

When dividing or multiplying both sides by a negative number, the inequality direction must flip. This is the defining rule for inequalities that distinguishes them from equations.

flip-on-negative Dividing or multiplying by a negative number reverses the direction of the inequality. Forgetting to flip is the most common error.

Example

Divide by a negative number and flip the inequality sign.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

−3x≤12- 3x \le 12
Watch the negative coefficient -3.-3x ≤ 12negative coefficient -3divisor will be -3abs value 3sign flip--interval--Start: -3x ≤ 12

Step 2 — Divide by negative

Use divisor -3 on the right side 12.

x≤12/-3x \le 12 / \hl{-3}
Divide both sides by -3.-3x ≤ 12negative coefficient -3x ≤ 12 / -3dividing by -3sign flip--interval--Start: -3x ≤ 12

Step 3 — Flip the sign

Divide by -3, so flip <= to >=.

≤→≥\le \rightarrow \hlmath{\ge}
Flip the sign: ≤ -> ≥.-3x ≤ 12negative coefficient -3x ≤ 12 / -3dividing by -3≤ -> ≥x ≥ -4interval[-4, ∞)Start: -3x ≤ 12

Step 4 — Solution

After the flip, the solution is x >= -4.

x≥-4x \ge \hl{-4}
x ≥ -4-7-6-5-4-3-2-1closed circle

Step 5 — Check a point

Check -3: -3 >= -4, so it is on the shaded side.

-3≥−4 shaded\hl{-3} \ge -4 \ \mathrm{shaded}
x ≥ -4-7-6-5-4-3-2-1closed circle

Step 6 — Interval

Write the interval starting at -4.

x∈[-4,∞)x\in[ \hl{-4} ,\infty)
x ≥ -4-7-6-5-4-3-2-1closed circle