Solving Linear Inequalities

Solving inequalities: almost like equations

Solving a linear inequality uses the same steps as solving an equation — add, subtract, multiply, divide — with one critical difference: multiplying or dividing by a negative number FLIPS the inequality sign.

Step-by-step: Solve $2x - 5 > 7$

Step 1 — Add 5 to both sides:

$2x - 5 + 5 > 7 + 5$ $2x > 12$

Step 2 — Divide both sides by 2 (positive, so inequality stays the same):

$x > 6$

Step 3 — Graph on a number line. Place an open circle at 6 (because $>$, not $\geq$) and shade to the right (all values greater than 6).

Step 4 — Interval notation: $(6, \infty)$.

Step 5 — Check with a test point. Try $x = 8$: $2(8) - 5 = 11 > 7$ ✓. Try $x = 4$: $2(4) - 5 = 3 \not> 7$ ✗.

The critical rule: flipping the inequality

If you multiply or divide by a negative number, the inequality reverses:

$-3x > 12 \implies x < -4 \quad \text{(divided by -3, flipped > to <)}$

Why? Because multiplying by a negative reverses order: $2 < 5$, but $-2 > -5$.

Graphing conventions

• Open circle (∘): strict inequality ($<$ or $>$) — boundary NOT included.
• Closed circle (●): non-strict ($\leq$ or $\geq$) — boundary IS included.
• Shade left for "less than," shade right for "greater than."

Try it in the visualization

Adjust the coefficient, constant, and right side. The number line shows the solution with proper open/closed circle and shading direction. A test point verifies whether it satisfies the inequality.