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C2.6 Representing and Interpreting Inequalities

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01 — Explanation

How Is an Inequality Different From an Equation?

An equation like $x=4$ has exactly one solution. An inequality like $x>4$ has infinitely many — $4.1$ , $5$ , $100$ , anything strictly greater than $4$ . Inequalities describe a whole range of acceptable values rather than a single answer, which is exactly why real-world limits (speed limits, weight limits, minimum ages) are written as inequalities rather than equations.

02 — Explanation

Which Circle Do I Draw — Open or Filled?

The circle tells you whether the boundary number itself counts as a solution:

Symbol	Meaning	Boundary Included?	Circle
$<$	strictly less than	No	Open $\circ$
$>$	strictly greater than	No	Open $\circ$
$\leqslant$	less than or equal to	Yes	Filled $\bullet$
$\geqslant$	greater than or equal to	Yes	Filled $\bullet$

Exam Tip

Read the symbol as a mouth — the "wide open" end of $<$ or $>$ faces the bigger number, but it's the strictness (no line under it) that tells you to use an open circle, not the direction it points.

03 — Explanation

How Do I Draw a Combined Inequality on a Number Line?

Show $-3 \leqslant x < 1$ on a number line:

◄──●───────○──►
  -3        1

At $x=-3$ : the symbol is $\leqslant$ (inclusive), so draw a filled circle.
At $x=1$ : the symbol is $<$ (strict), so draw an open circle.
Join the two circles with a solid line, since every value between $-3$ and $1$ satisfies the inequality.

04 — Explanation

How Do I Read an Inequality Straight Off a Diagram?

Work backwards from the diagram: an open circle means that value is excluded (strict inequality); the direction of the arrow (or the solid line, for a combined inequality) tells you whether values are greater or less than the boundary.

A diagram shows an open circle at $-2$ with an arrow pointing right (towards larger numbers). Since the circle is open, the symbol is strict, and since the arrow points towards increasing values:

x > -2

05 — Explanation

How Do I Write an Inequality From a Description in Words?

Identify the boundary value first, then decide whether that exact value is allowed — words like "at most", "no more than" or "maximum" include the boundary ( $\leqslant$ ), while "more than" or "exceeds" exclude it ( $>$ ).

A lift has a safe working load of $500\text{ kg}$ . Write an inequality for the total mass $m$ that can safely be carried.

Since $500\text{ kg}$ itself is still safe, the boundary is included:

m \leqslant 500

Exam Tip

"At least" ( $\geqslant$ ) and "at most" ( $\leqslant$ ) both include their boundary value — it's easy to mistakenly treat them as strict inequalities under exam pressure.

06 — Explanation

How Do I List Whole-Number Solutions to an Inequality?

Check each end of the inequality separately for whether it's included, then list every integer in between.

Given that $x$ is an integer, list all values satisfying $-3 < x \leqslant 2$ :

$x>-3$ is strict, so $-3$ itself is not included.
$x \leqslant 2$ is inclusive, so $2$ is included.

\{-2,\ -1,\ 0,\ 1,\ 2\}

Exam Tip

It's easy to accidentally include $-3$ in the list above out of habit — always double-check the symbol touching each boundary number before writing the final list.

07 — Explanation

Exam-Style Question

$x$ is an integer such that $-4 \leqslant x < 0$ . Represent this inequality on a number line, and list all possible values of $x$ .

Solution

◄──●───────────○──►
  -4            0

At $x=-4$ : symbol is $\leqslant$ , so a filled circle. At $x=0$ : symbol is $<$ , so an open circle.

Listing the integers from $-4$ up to (but not including) $0$ :

\{-4,\ -3,\ -2,\ -1\}