Inequality calculator: linear and quadratic

Example: $3x + 5 < 20$. Subtract 5: $3x < 15$. Divide by 3: $x < 5$.

With a sign flip: $2x + 1 \geq 5x - 8$. Subtract $5x$: $-3x + 1 \geq -8$. Subtract 1: $-3x \geq -9$. Divide by $-3$ and reverse the sign: $x \leq 3$.

1. Solve the equation $= 0$ to find the critical values.
2. Sketch the graph: a positive $x^2$ term gives a ∪ shape.
3. Read off where the graph is below the axis (for $<$) or above it (for $>$).

Example: $x^2 - 5x + 6 < 0$. Roots $x = 2$ and $x = 3$. The ∪-shaped graph is below the axis between them: $2 < x < 3$.

Example: $x^2 - 9 > 0$. Roots $\pm 3$; above the axis outside them: $x < -3$ or $x > 3$.

Integer solutions: for $-2 < x \leq 3$, the whole numbers are $-1, 0, 1, 2, 3$.

For $-1 < 2x + 3 \leq 9$, do the same thing to all three parts: subtract 3 to get $-4 < 2x \leq 6$, then divide by 2: $-2 < x \leq 3$. Our inequalities questions include these and number-line questions.

• Forgetting to flip the sign when dividing by a negative.
• For quadratics, writing $x < 2$ and $x < 3$ instead of reading the graph.
• Using a filled circle for $<$ on a number line.

See our guide to linear inequalities, and solve the quadratic equation part with the quadratic equation solver.

• Keep the $x$ term positive where possible (move it to the side where it stays positive) to avoid sign flips.
• For quadratics, always include the sketch: it's where most marks come from.
• Use "and" for a region between two values, written as one double inequality; use "or" for two separate regions.