Sec 2 Linear Inequalities: Number Lines & Worked Examples

Secondary 2 G3 Mathematics · Study guide

Connect inequality rules to number-line solution sets, handle negative divisors and translate practical limits into valid whole-number answers.

Concept → worked examples → practice12 min read · practice linked below

An inequality describes a set of possible values

An equation asks when two expressions are equal. An inequality compares which expression is greater or smaller, so its answer is usually a range of values. The boundary and whether that boundary is included both matter.

Before you begin: know signed-number arithmetic, fraction operations, expanding brackets and solving linear equations. Keep the inequality sign in every line of your working.

Read the comparison carefully
Symbol	Meaning	Boundary included?
$<$	Less than	No
$>$	Greater than	No
$\le$	At most; no more than	Yes
$\ge$	At least; no less than	Yes

Add or subtract the same quantity on both sides without changing the comparison. Multiplying or dividing both sides by a positive number keeps the sign; multiplying or dividing by a negative number reverses it. For example, $2<5$ , but $-2>-5$ .

Subtracting a negative number does not itself reverse the sign. The reversal comes from multiplying or dividing by a negative number.

Read the answer on a number line

Worked example 1 — an included boundary

Solve $x+4\le2$ and represent the answer on a number line.

Subtract 4 from both sides: $x\le2-4$ , so $\boxed{x\le-2}$ . The answer contains $-2$ itself and every number smaller than $-2$ .

A filled circle includes the boundary. For

x<-2

, the ray would point the same way but the circle would be hollow.

Check: $x=-2$ gives $2\le2$ , which is true; $x=-3$ gives $1\le2$ , also true. A value outside the range, $x=0$ , gives $4\le2$ , which is false.

A negative divisor reverses the direction

Worked example 2 — a strict negative boundary

Solve $5-3x>14$ , then state the greatest integer solution.

Subtract 5 from both sides to isolate the term containing $x$ . Then divide both sides by $-3$ , reversing $>$ to $<$ :

\begin{aligned}-3x>9\\x<-3.\end{aligned}

The greatest integer less than $-3$ is $-4$ . The boundary $-3$ is excluded because the inequality is strict. More negative integers, such as $-5$ , also work, but they are smaller than $-4$ .

Check: $x=-4$ gives $5-3(-4)=17>14$ . At the boundary $x=-3$ , the left side is 14, so $14>14$ is false. A number line would use a hollow circle at $-3$ and a ray pointing left.

A quick direction check is to substitute one value on the chosen side of the boundary. This can reveal a missed sign reversal even if the boundary calculation was correct.

Fractions and brackets: keep every term

Worked example 3 — clear fractions, then select special values

Solve $\dfrac{2x-1}{3}-\dfrac{x+2}{4}<2$ . State the greatest integer, prime number and perfect square satisfying it.

The lowest common multiple of 3 and 4 is 12. Multiply every term by this positive number, so the inequality direction stays the same:

\begin{aligned}4(2x-1)-3(x+2)<24\\8x-4-3x-6<24\\5x-10<24\\5x<34\\x<\frac{34}{5}=6.8.\end{aligned}

The greatest permitted integer is 6. The greatest prime below 6.8 is 5, and the greatest perfect square below 6.8 is 4. These are different questions: an integer does not have to be prime or square.

Check: at $x=6$ , $\frac{11}{3}-\frac84=\frac53<2$ . At $x=7$ , $\frac{13}{3}-\frac94=\frac{25}{12}>2$ , so 7 is outside the range. At $x=\frac{34}{5}$ , the left side is exactly 2, so the boundary is excluded.

For a question asking for a smallest or greatest integer, solve the inequality first. Then choose an integer in the permitted direction. Ordinary rounding is unreliable: if $x>6.1$ , the smallest integer is 7; if $x<-2.7$ , the greatest integer is $-3$ .

Translate a limit into an inequality

Worked example 4 — the largest affordable whole number of hours

A bicycle hire shop charges a fixed fee of $8 plus $3.50 per hour. A customer has $30 and hires the bicycle for a whole number of hours. What is the greatest number of hours they can afford?

Let $h$ be the number of hours. The total cost is $8+3.50h$ dollars. Spending no more than $30 means:

\begin{aligned}8+3.50h&\le30\\3.50h&\le22\\h&\le\frac{22}{3.50}=\frac{44}{7}=6\frac27.\end{aligned}

The context requires a non-negative whole number of hours. The greatest whole number at most $6\frac27$ is 6, so the customer can afford 6 hours.

Check both sides of the decision: 6 hours costs $8+3.50(6)=\$29$ , within the budget. Seven hours costs $8+3.50(7)=\$32.50$ , which exceeds it. Rounding the bound up to 7 would give an unaffordable answer.

Explore the effect of a negative or zero coefficient

Solve $ax+b$ compared with $c$

Change the coefficient, constants and comparison. Watch the direction of the solution ray and the boundary circle. When $a=0$ , the comparison contains no $x$ , so it is either true for every real number or false for every real number.

The default comparison is $-2x+3\le7$ . Subtract 3, then divide by $-2$ and reverse the sign: $x\ge-2$ .

−2x + 3 ≤ 7

A hollow circle excludes the boundary; a filled circle includes it. The ray continues beyond the displayed number line.

Subtract 3: −2x ≤ 4. Divide by −2 and reverse the sign: x ≥ −2. The boundary is included.

Common mistakes and a reliable final check

Reversing the sign for addition or subtraction: reverse it only when multiplying or dividing by a negative number.
Losing brackets after clearing fractions: $-3(x+2)=-3x-6$ , not $-3x+6$ .
Using a filled circle for a strict inequality: equality is allowed only with $\le$ or $\ge$ .
Choosing the wrong integer near a negative boundary: place the values on a number line; farther left means smaller.
Giving only a boundary: write $x<6.8$ , not just 6.8. The sign is part of the answer.

Check the boundary in the original inequality, test one value inside the range and one outside it, and check any whole-number or unit restrictions in the question. A prime number is an integer greater than 1 with exactly two positive factors, so 1 is not prime.

Practise ranges, number lines and limits

Use the Sec 2 linear inequalities practice bank for questions and worked solutions. Start with sign reversal, then try fractional inequalities, special-number answers and word problems. Always connect your algebraic answer to what the number line or context means.