Solving quadratic inequalities with number lines

A quadratic inequality asks which values of (x) make one quadratic expression greater than, less than, or equal to another quantity. Unlike a quadratic equation, which usually has one or two solutions, an inequality can produce whole intervals of values. A number line shows those intervals clearly.

The key idea is to find where the quadratic crosses the horizontal axis, then track how its sign changes. The roots become boundary points, while the direction in which the parabola opens determines which regions are shaded.

Australian schools increasingly use algebraic reasoning beyond routine equations, as outlined in ACARA’s Australian Curriculum Mathematics Version 9. Number-line diagrams are especially valuable because they connect symbolic algebra with a visual representation that can be read without calculating several test values.

Many Australian students complete homework during the January summer term or while travelling on buses and trains around cities such as Sydney and Melbourne. Keeping a clear number line helps separate a difficult algebraic step from the final interpretation. Australian textbooks also use a decimal point rather than a comma, so approximate roots should follow that convention.

Online homework assistance, mathematical tools and step-by-step solutions form a substantial part of Australia’s education market. These services can provide useful checks, but students should understand the method rather than rely on an answer alone. Any platform collecting student information must also meet the relevant requirements of the Privacy Act 1988 and the Australian Privacy Principles.

Understand what the inequality requires

An inequality uses symbols such as (<), (>), (\leq) and (\geq) rather than an equals sign. These symbols matter when the answer is drawn: a solid dot represents an included endpoint, while an open circle represents an excluded endpoint.

Before solving, put the quadratic on one side of the comparison. For example,

[ x^2-5x+6\leq 0 ]

is already in a suitable form because zero is on the right. If both sides contain terms, move everything to one side and use the same inequality symbol throughout.

A dependable method is to find the critical values, determine the sign of the quadratic between them, and then convert the result into interval notation. The critical values are the values of (x) for which the quadratic equals zero.

Find the critical values

Start by replacing the inequality sign with an equals sign. Solve the resulting quadratic equation using factorisation, the quadratic formula or completing the square. For example,

[ x^2-5x+6=0 ]

factors as

[ (x-2)(x-3)=0. ]

The roots are (x=2) and (x=3), so these values divide the number line into three regions.

The roots are not always whole numbers. The discriminant (b^2-4ac) gives useful information before any calculation: a positive value produces two real roots, zero produces one repeated root, and a negative value produces no real roots. If there are no crossings, the parabola keeps the same sign throughout the visible number line.

Turn signs into a number-line diagram

A sign chart records what happens on each side of the roots. Its columns can show (x), the critical values, the value of the quadratic and its sign. This prevents gaps in reasoning, especially when several expressions must be combined.

Whether the parabola opens upwards or downwards controls the shading. If (a>0), the arms point upwards, so the quadratic is positive outside its roots and negative between them. If (a<0), the pattern reverses.

The original inequality determines the final selection. For an upward-opening parabola, (f(x)>0) selects the outside regions, while (f(x)<0) selects the region between the roots. Equality includes the boundary points unless the inequality is strict.

Solve a complete example

Consider

[ x^2-5x+6\leq 0. ]

Factorising gives ((x-2)(x-3)\leq 0), with critical values (2) and (3). Since the leading coefficient is positive, the parabola opens upwards. It is negative between the roots and zero at (2) and (3).

Therefore, the solution lies between the two critical values. Because the inequality includes equality, both endpoints are included. In interval notation, the solution set is

[ [2,3]. ]

On the number line, place solid dots at (2) and (3), then shade the section joining them. The shaded set represents every real number (x) for which the quadratic is zero or negative.

Use the vertex when roots are awkward

Some quadratics do not factor neatly into integers. In that case, completing the square connects the problem directly with vertex form. Consider

[ x^2-4x-3\leq 0. ]

Completing the square gives

[ (x-2)^2-7\leq 0, ]

so

[ (x-2)^2\leq 7. ]

The roots are (2-\sqrt7) and (2+\sqrt7), approximately (-0.65) and (4.65). The parabola opens upwards, so the values satisfying the inequality lie between these roots.

The exact interval is

[ [2-\sqrt7,;2+\sqrt7]. ]

Exact forms are preferable when a marking scheme requires them, although decimal approximations can help when drawing the number line. Either version must show that the endpoints are included.

Prevent common errors

One frequent mistake is to assume that every quadratic is negative between its roots. That is true only when the parabola opens upwards. A downward-opening parabola has the opposite sign pattern, so the inequality may select the two outside regions instead.

Another error is dividing an inequality by an expression involving (x) without checking its sign. Dividing by a negative quantity reverses the inequality, and dividing by a quantity that might be zero requires separate consideration. Moving every term to one side and using a sign chart is usually safer.

Finally, distinguish the union symbol (\cup) from the intersection symbol (\cap). A union combines separate solution intervals, while an intersection keeps only values belonging to both. The number line should always match the final interval notation.

Check the method before recording the answer

Before shading, confirm that the graph’s direction and the inequality symbol agree.

• Identify the two real roots or the repeated root.
• Place each critical value in ascending order.
• Decide whether the parabola opens upwards or downwards.
• Test one value inside each resulting interval.

After shading, check every boundary and the region beyond it.

• Use an open circle for (<) or (>).
• Use a solid dot for (\leq) or (\geq).
• Write separate intervals with (\cup) when needed.
• Confirm that one test value satisfies the original inequality.

Your next step is to take an unused quadratic inequality, find its critical values and draw one labelled number line with the correct endpoints and shading.