A22 - Linear inequalities and number lines

Ranges satisfying a linear inequality

A minibus costs £40 plus £8 per passenger. With £100 to spend, can you afford eight passengers? An inequality will give the whole range of affordable counts, rather than just one trial.

An inequality sign compares values. The difference between strict and inclusive signs matters at the boundary.

Unless a domain is stated, do not assume the unknown is an integer. For example, $x<4$ allows 3.9, zero and negative numbers.

Change only $<$ to $\leq$. The algebra is unchanged, but the endpoint is now included.

Variables on both sides use the same collection method. For $4x+1\geq2x+9$, subtracting $2x$ and 1 gives $2x\geq8$, hence $x\geq4$.

Brackets and fractions also follow the equation methods. Expanding $2(x-1)\leq x+3$ gives $2x-2\leq x+3$, then $x\leq5$. Multiplying $(x+1)/3>2$ by positive 3 gives $x+1>6$, hence $x>5$.

Change only the right-hand number from Worked example 2.

Negative multiplication reverses order

Multiplying or dividing both sides by a negative number reverses their order.

Start with $2<5$. Multiplying by $-1$ gives $-2>-5$: the points have reflected across zero. The greater original number becomes the smaller negative number. Every negative multiplier includes this reversal.

For example, $-3x<12$ gives $x>-4$ after dividing by $-3$. A permitted input $x=0$ gives $0<12$. A wrong-side input $x=-5$ gives $15<12$, false.

Change only the comparison to a strict one. The endpoint will be excluded.

You can avoid negative division by moving the variable instead. From $7-2x\leq1$, add $2x$, subtract 1 and divide by 2:

This is the same range as $x\geq3$.

Do not divide by an unknown unless its sign is known. In $xy>3x$, the direction after dividing by $x$ would depend on whether $x$ is positive or negative; division is impossible if $x=0$.

Now change only the right-hand number from Worked example 4.

Open and closed number-line endpoints

A number line shows the allowed values and whether each endpoint belongs to the solution.

[DIAGRAM: asset_name: Strict and inclusive number-line solutions; asset_slug: 1ma1_m32_number_lines; caption: Endpoint style records inclusion, while the shaded direction records the allowed values.; description: Three horizontal number lines show x > 2 with an open circle and a right arrow, x ≥ 2 with a filled circle and a right arrow, and −1 < x ≤ 3 with an open circle at −1, filled circle at 3 and a highlighted segment between them.]