A22 - Solving linear inequalities

Reading a solution set on a number line

A lift can carry at most $600\,\mathrm{kg}$. Its passengers and luggage already have mass $420\,\mathrm{kg}$. How much more mass can go in? Is the boundary amount allowed? An inequality gives both the limit and the answer to that last question.

For $x>2$, numbers such as $2.1$, $3$ and $100$ work, but $2$ itself does not. For $x\geqslant2$, the boundary $2$ works as well. Both include decimals and fractions unless a question restricts $x$ to integers.

On a number line, use an open circle for an excluded boundary and a filled circle for an included boundary. An arrow shows that the solution continues without an end in that direction. A segment between two boundary circles shows a bounded range.

[DIAGRAM: asset_name: Open and filled boundary circles; asset_slug: aqa8300_a12_number_lines; caption: The circle decides whether an endpoint belongs to the solution; the line decides which values belong; description: Three number lines. Top x greater than 2 has an open circle at 2 and a right arrow. Middle x greater than or equal to 2 has a filled circle at 2 and a right arrow. Bottom minus 1 less than x less than or equal to 3 has an open circle at minus 1, a filled circle at 3 and a coloured segment between them.]

The bottom line means $-1<x$ and $x\leqslant3$, written together as $-1<x\leqslant3$. The numbers are in increasing order. These number-line conventions are not on the formulae sheet: learn them.

There are infinitely many solutions between $2$ and $3$, although there is no integer strictly between them. For example, $2.5$, $2.25$, $2.125$ and the continuing pattern all lie there. Between any two different numbers you can choose their midpoint, so you cannot run out of numbers to insert.

Check an excluded endpoint carefully in the following question.

Solving with addition and positive division

You can add or subtract the same number on both sides of an inequality without changing its direction. Multiplying or dividing by the same positive number also keeps the direction. These operations preserve the order of the values on a number line.

Start with $2<5$. Adding $4$ gives $6<9$. Dividing both original numbers by $2$ gives $1<2.5$. Neither operation reverses which is smaller.

These non-calculator examples undo an addition, then a positive multiplication. Keep an inequality sign in every line.

Change only the sign to an inclusive one. The boundary calculation stays the same, but the boundary is now allowed.

Solving the corresponding equation finds the boundary. It does not tell you which side of it is allowed unless you retain the inequality.

Use the same steps with a new coefficient and boundary.