2.6 Linear and quadratic inequalities, and set notation — Learn

Prerequisites: solving linear equations; factorising and sketching a parabola from 2.3; the number line.

By the end of this section you can: solve a linear inequality and remember to reverse the sign when you multiply or divide by a negative number; solve a quadratic inequality by comparing the parabola with the $x$-axis; and write the solution as an interval, as a union, or in words.

1. An inequality has a range of answers

An equation has a list of answers. An inequality has a range of answers, and the answer is a set of numbers. Solve an inequality the way you solve an equation, with one extra rule.

Adding or subtracting the same number keeps the direction of the inequality.

Multiplying or dividing by a negative number reverses the direction.

For example, $-2x < 6$ gives $x > -3$, because dividing by $-2$ turns $<$ into $>$. Check with $x = 0$: the original is $0 < 6$, which is true, and $0 > -3$ is also true.

2. Set notation

A solution set can be written in several ways. The interval notation is the one used most at A level.

A round bracket excludes the end point; a square bracket includes it. The symbols $\infty$ and $-\infty$ always take a round bracket, because infinity is not a number you can reach.

Example 1. Solve $2x - 3 < 5x + 6$.

Subtract $2x$ from both sides:

Subtract $6$:

Divide by $3$, a positive number, so the direction does not change:

The solution set is $x \in (-3, \infty)$.

3. Quadratic inequalities

A quadratic inequality asks where a parabola is above or below the $x$-axis. The method has three steps.

1. Solve the related equation to find the roots.
2. Sketch the parabola: it opens upwards when the coefficient of $x^2$ is positive.
3. Read the sign from the sketch.

Example 2. Solve $x^2 - x - 6 < 0$.

First solve $x^2 - x - 6 = 0$. Factorise:

so the roots are $x = -2$ and $x = 3$.

The parabola $y = x^2 - x - 6$ opens upwards, so it is below the $x$-axis between the roots and above it outside them. The inequality asks for $< 0$:

So the solution set is $x \in (-2, 3)$.

Example 3. Solve $x^2 + 2x - 8 > 0$.

Factorise the related equation:

so the roots are $x = -4$ and $x = 2$. The parabola opens upwards and the inequality asks for $> 0$, which is outside the roots:

In set notation,

The symbol $\cup$ means union: the answer is everything in the first interval, together with everything in the second. When the answer has two pieces, write it as a union.

Example 4. Solve $x^2 - 6x + 9 \leq 0$.

Factorise:

A square is never negative, so $(x - 3)^2 \leq 0$ only when $(x - 3)^2 = 0$. The only solution is $x = 3$.

4. Inequalities in two variables

An inequality such as $y > x + 1$ describes a region of the plane, not a range of $x$.

• Draw the boundary line $y = x + 1$.
• Use a dashed line for a strict inequality ($<$, $>$) and a solid line for $\leq$ or $\geq$.
• Shade the region that satisfies the inequality. Test one point, such as $(0, 0)$, to decide which side.

For $y > x + 1$, the point $(0, 0)$ gives $0 > 1$, which is false, so the region above the line is the solution and $(0, 0)$ is not in it.

5. Try it: reading the solution set

Predict first. For the roots $x = -2$ and $x = 3$, the inequality $x^2 - x - 6 > 0$ is solved outside the roots and $x^2 - x - 6 < 0$ is solved between them. Which one gives a single interval, and which gives a union of two intervals? Write your prediction down before you touch the controls.

Test. Move the two roots and choose the inequality. The number line marks the solution set, with an open circle at a value the set leaves out and a filled circle at a value it includes.

Explain. Keep the roots fixed and switch between $f(x) > 0$ and $f(x) < 0$. Why does one inequality give a single interval and the other give two separate pieces? Write two sentences referring to where the parabola sits above and below the $x$-axis.

Watch out. Two errors are common.

Forgetting to reverse the sign. Dividing $-2x > 8$ by $-2$ gives $x < -4$, not $x > -4$. Test one value to check the direction.

Choosing the wrong side for a quadratic. $x^2 - x - 6 < 0$ is solved between the roots, but $x^2 - x - 6 > 0$ is solved outside them. Sketch the parabola every time.

Re-check: pick a number from your solution set, substitute it into the original inequality, and confirm that it is true.

Progress check

Return to the chapter problem. The arch is $y = 4 - \frac{x^2}{4}$ and the lorry roof is $3.2$ m above the road.

1. Write an inequality for the positions $x$ where the arch is at least as high as the lorry roof.
2. Solve the inequality $4 - \frac{x^2}{4} \geq 3.2$ and give the solution as an interval.
3. The arch is wide enough at roof height for a lorry of what width? Give your answer in exact form and to two decimal places.
4. The lorry is $2.5$ m wide. Use your interval to decide whether the lorry fits, and write one sentence of conclusion.

You should now be able to

• solve a linear inequality and reverse the sign when dividing by a negative;
• solve a quadratic inequality by using the roots and the shape of the parabola;
• write a solution set as an interval or a union of intervals;
• decide whether an endpoint is included from the inequality symbol;
• describe the region for an inequality in two variables.