Year 10 Maths - 10A Linear Relations

Linear Inequalities

Q: What is the difference between an open and a closed dot?

An open dot means the end number is not included, for less than or greater than. A closed dot means it is included, for less than or equal to and greater than or equal to.

Q: How do you write at least and at most as inequalities?

At least means greater than or equal to, and at most means less than or equal to. More than and fewer than use the strict signs, greater than and less than.

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Theory

To solve a linear inequality, use the same steps as for an equation, but reverse the inequality sign whenever you multiply or divide by a negative number. This Year 10A maths page covers brackets, fractions, double inequalities, graphing and reading solutions on a number line, integer solutions, worded problems and unknown constants.

An inequality compares two expressions with $\lt $ (less than), $\gt $ (greater than), $\le$ (less than or equal to) or $\ge$ (greater than or equal to). Its solution is usually a whole set of numbers, such as $x \gt 3$, shown on a number line.

You can add or subtract any number on both sides, and multiply or divide both sides by a positive number, without changing the sign. If you multiply or divide by a negative number, the order of the numbers is reversed, so the sign must be reversed too: from $-2x \lt 6$ you get $x \gt -3$.

Swapping the two sides also reverses the sign: $-3 \lt x$ means the same as $x \gt -3$. A double inequality such as $-1 \lt 2x+3 \le 7$ is two inequalities at once; apply each step to all three parts.

To read a graph on a number line, look at the boundary number, the dot (open: not included; closed: included) and the direction of the shading. If the pronumeral must be a whole number, list the integers in the shaded part, including an end point only if its dot is closed.

Use a closed dot for ≤ or ≥ and an open dot for < or >. Shade the side that works.

Multiplying or dividing by a negative number reverses the order, so the inequality sign must be reversed.

For any numbers $a$, $b$ and $c$:

Step	Rule	Sign
Add or subtract $c$	$a\lt b \Rightarrow a\pm c\lt b\pm c$	stays
Multiply or divide by $c\gt 0$	$a\lt b \Rightarrow ca\lt cb$	stays
Multiply or divide by $c\lt 0$	$a\lt b \Rightarrow ca\gt cb$	reverses
Swap the sides	$a\lt b \Leftrightarrow b\gt a$	reverses

a < b, c < 0 \Rightarrow c a > c b

Dividing by a letter. Before dividing by an expression such as \(b-3\), decide its sign from the information given. If \(b \lt  3\), then \(b-3\) is negative and the sign reverses: \((b-3)x \gt  5\) gives \(x \lt  \dfrac{5}{b-3}\).

Which inequality matches the graph? Solve each option and compare its solution with the graph. The boundary number, the type of dot and the direction of the shading must all match.

Working backwards from a graph. If a graph shows \(-2 \lt x \le 3\) as the solution of \(a \lt 2x + 1 \le b\), build \(2x + 1\) from \(x\): \(-4 \lt 2x \le 6\), then \(-3 \lt 2x + 1 \le 7\). So \(a = -3\) and \(b = 7\).

How to solve a linear inequality

Clear fractions by multiplying every term by the LCD (a positive number, so the sign stays).
Expand any brackets, taking care with a negative in front.
Collect the pronumeral terms on one side and the numbers on the other. Collecting them on the side where the coefficient stays positive avoids dividing by a negative.
Divide by the coefficient. If it is negative, reverse the sign.
Write the answer with the pronumeral first, such as $x \le -2$, and graph it if asked.
Check with one number inside the solution and one outside.

Other question types

Two inequalities together: solve each one, then find the numbers that satisfy both. If they do not overlap, such as $x \gt 5$ and $x \lt 2$, there is no solution.
Integer solutions: list the whole numbers in the solution set, taking care with the end points (Example 2). For the largest integer with $n \lt 27$, the answer is $26$.
Worded problems: "at least" means $\ge$, "at most" means $\le$, "more than" means $\gt $. A mean of at least $65$ over five tests means the total is at least $5\times 65$.
Whole-number answers: if $n$ must be a whole number, round in the direction that keeps the inequality true (Example 3).
Unknown constants: solve for $x$ in terms of the constant, then apply the condition (Example 4). If the solution is given on a graph, read it off and match it, or build the expression from $x$ (see Formulas).

Example 1 — Reversing the sign

Solve and graph:
(i) $7-2(x+4)\ge 3x+9$
(ii) $\dfrac{1-x}{3}\gt \dfrac{x+5}{4}$

Solution

(i) Expand, then collect the $x$ terms on the side where they stay positive.

$7-2x-8$	$\ge$	$3x+9$
$-1-9$	$\ge$	$3x+2x$
$-10$	$\ge$	$5x$
$-2$	$\ge$	$x$
$x$	$\le$	$\boldsymbol{-2}$

7 - 2 (x + 4) \geq 3 x + 9 \Rightarrow x \leq - 2

(ii) Multiply both sides by $12$. It is positive, so the sign stays.

$4(1-x)$	$\gt $	$3(x+5)$
$4-4x$	$\gt $	$3x+15$
$-7x$	$\gt $	$11$
$x$	$\lt $	$\boldsymbol{-\dfrac{11}{7}}$

- 7 x > 11 \Rightarrow x < - \frac{11}{7}

Dividing by $-7$ reverses $\gt $ to $\lt $.

Example 2 — A double inequality

Solve $1\lt 7-2x\le 13$ and list the integers that satisfy it.

Solution

Do the same thing to all three parts. Subtract $7$, then divide by $-2$ and reverse both signs.

$1-7$	$\lt $	$-2x\le 13-7$
$-6$	$\lt $	$-2x\le 6$
$3$	$\gt $	$x\ge -3$

Write it with the smaller number first:

$-3$

$\le$

$\boldsymbol{x\lt 3}$

- 3 \leq x < 3

$-3$ is included and $3$ is not. The integers are $\mathbf{-3,\ -2,\ -1,\ 0,\ 1,\ 2}$.

Example 3 — Making a profit

A café spends $\$480$ a day on rent and wages. Each coffee costs $\$1.30$ to make and sells for $\$4.50$. How many coffees must it sell in a day to make a profit?

Solution

Let $n$ be the number of coffees sold. Profit means income is more than costs.

$4.50n$	$\gt $	$480+1.30n$
$3.20n$	$\gt $	$480$
$n$	$\gt $	$150$

4.50 n > 480 + 1.30 n \Rightarrow n > 150

At $150$ coffees the café only breaks even ($\$675$ in, $\$675$ out). $n$ is a whole number greater than $150$, so it must sell at least $\mathbf{151}$ coffees.

If the answer had been $n \gt 150.4$, you would still round up to $151$: rounding down would not meet the condition.

Example 4 — An unknown constant

Find the values of $m$ for which the solution of $4x+m = 2x-6$ is negative.

Solution

Solve the equation for $x$ in terms of $m$ first.

$2x$	$=$	$-6-m$
$x$	$=$	$\dfrac{-6-m}{2}$

Now require $x \lt 0$. Multiplying by $2$ keeps the sign.

$\dfrac{-6-m}{2}$	$\lt $	$0$
$-6-m$	$\lt $	$0$
$-m$	$\lt $	$6$
$m$	$\gt $	$\boldsymbol{-6}$

\frac{- 6 - m}{2} < 0 \Rightarrow m > - 6

Check: $m = 0$ gives $x = -3$, which is negative. $m = -10$ gives $x = 2$, which is not.

Common pitfalls

Reverse the sign for negatives only. From $-3x \gt 12$, dividing by $-3$ gives $x \lt -4$. But from $3x \gt -12$ the sign stays: $x \gt -4$. A negative answer does not reverse anything; a negative divisor does.

Reverse both signs in a double inequality. Dividing $-6 \lt -3x \le 9$ by $-3$ gives $2 \gt x \ge -3$. Then rewrite it smallest first: $-3 \le x \lt 2$.

Watch the dots on a number line. $\lt $ and $\gt $ get an open dot (the end point is not included); $\le$ and $\ge$ get a closed dot.

Round the right way. If $n \gt 52.3$ and $n$ is a whole number, the least value is $53$. If $n \lt 18$, the largest whole number is $17$, not $18$.

Check the sign of a letter before dividing. You cannot divide by $b-3$ without knowing whether it is positive or negative. The question must tell you.

Frequently asked questions

Why do you flip the inequality sign when dividing by a negative?

Multiplying or dividing by a negative number reflects the numbers in zero on the number line, which reverses their order. For example, $2 \lt 5$ but $-2 \gt -5$. The sign is reversed to keep the statement true.

Do you flip the sign when subtracting a negative?

No. Adding or subtracting any number, positive or negative, never changes the sign. Only multiplying or dividing by a negative number, or swapping the two sides, reverses it.

How do you solve a double inequality?

Treat all three parts at once. Whatever you add, subtract, multiply or divide, do it to the left, middle and right. If you divide by a negative, reverse both signs, then rewrite the answer with the smaller number on the left.

What is the difference between an open and a closed dot?

An open dot means the end number is not included, for $\lt $ or $\gt $. A closed dot means it is included, for $\le$ or $\ge$.

How do you write at least and at most as inequalities?

"At least" means $\ge$ and "at most" means $\le$. "More than" and "fewer than" use the strict signs $\gt $ and $\lt $.

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Graphing Straight Lines

\(7-2x-8\)	\(\ge\)	\(3x+9\)
\(-1-9\)	\(\ge\)	\(3x+2x\)
\(-10\)	\(\ge\)	\(5x\)
\(-2\)	\(\ge\)	\(x\)
\(x\)	\(\le\)	\(\boldsymbol{-2}\)

\(4(1-x)\)	\(\gt \)	\(3(x+5)\)
\(4-4x\)	\(\gt \)	\(3x+15\)
\(-7x\)	\(\gt \)	\(11\)
\(x\)	\(\lt \)	\(\boldsymbol{-\dfrac{11}{7}}\)

\(1-7\)	\(\lt \)	\(-2x\le 13-7\)
\(-6\)	\(\lt \)	\(-2x\le 6\)
\(3\)	\(\gt \)	\(x\ge -3\)

\(4.50n\)	\(\gt \)	\(480+1.30n\)
\(3.20n\)	\(\gt \)	\(480\)
\(n\)	\(\gt \)	\(150\)

\(2x\)	\(=\)	\(-6-m\)
\(x\)	\(=\)	\(\dfrac{-6-m}{2}\)

Step	Rule	Sign
Add or subtract \(c\)	\(a\lt b \Rightarrow a\pm c\lt b\pm c\)	stays
Multiply or divide by \(c\gt 0\)	\(a\lt b \Rightarrow ca\lt cb\)	stays
Multiply or divide by \(c\lt 0\)	\(a\lt b \Rightarrow ca\gt cb\)	reverses
Swap the sides	\(a\lt b \Leftrightarrow b\gt a\)	reverses

\(\dfrac{-6-m}{2}\)	\(\lt \)	\(0\)
\(-6-m\)	\(\lt \)	\(0\)
\(-m\)	\(\lt \)	\(6\)
\(m\)	\(\gt \)	\(\boldsymbol{-6}\)

Linear Inequalities

📖 Theory

How to solve a linear inequality

Other question types

Common pitfalls

Frequently asked questions

Theory