YEAR 10 · LINEAR EQUATIONS AND INEQUALITIES · 18 MIN

One-variable inequalities

Solve inequalities using equality-preserving steps, reversing the direction when multiplying or dividing by a negative number.

In this lesson

Solve strict and inclusive inequalities
Represent solution sets on a number line

Before you start

Linear equations
Ordered numbers

01 / LEARN

Understand the idea.

An inequality describes a set

An inequality describes an allowed set of values. Adding or subtracting the same amount on both sides preserves its order. Multiplication or division by a positive number also preserves it. Keep the inequality symbol on every line as you isolate the variable.

Why a negative reverses direction

Multiplying or dividing by a negative reverses the order: $2 < 5$ becomes $- 2 > - 5$ after multiplication by $- 1$ . On a number line, reflection about zero reverses left and right, which explains the change of symbol.

A strict boundary uses an open circle; a boundary allowing equality uses a filled circle. Test an included value and the boundary in the original inequality to check the final set. Testing a few values checks working but does not replace algebraic solution.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Divide by a negative

Solve $- 3 x + 5 \leq 14$ .

Worked solution

1
Subtract five from both sides.
$- 3 x \leq 9$
Subtraction preserves the direction of an inequality.
2
Divide by negative three and reverse the symbol.
$x \geq - 3$
Negative division reverses the order on the number line.
3
Describe the solution set.
$x \geq - 3$
The boundary $- 3$ is included, so use a filled circle there and shade to the right.
4
Check the boundary and an interior value.
$\begin{aligned} x = - 3 : & - 3 (- 3) + 5 = 14 \leq 14 \\ x = 0 : & - 3 (0) + 5 = 5 \leq 14 \end{aligned}$
Equality is allowed at the boundary, and the value $0$ lies in the shaded set.

Answer

$x \geq - 3$

EXAMPLE 02

Brackets and both sides

Solve $2 (3 x - 1) < 4 x + 8$ .

Worked solution

1
Expand the left side.
$6 x - 2 < 4 x + 8$
The factor $2$ multiplies both terms inside the bracket.
2
Gather the variable terms and constants.
$\begin{aligned} 2 x - 2 & < 8 \\ 2 x & < 10 \end{aligned}$
Subtracting $4 x$ and adding $2$ preserve the inequality direction.
3
Divide by the positive coefficient.
$x < 5$
Division by positive $2$ preserves the strict inequality.
4
Interpret the strict endpoint.
$x = 5 : 2 (3 \times 5 - 1) = 4 \times 5 + 8 = 28$
The two sides are equal at $5$ , so that boundary is excluded; shade left of an open circle.

Answer

$x < 5$

03 / YOUR TURN

Try it yourself.

Check your understanding

Solve $7 - 2 x > 1$ .

Show answer & reasoning

$x < 3$

Subtract $7$ to get $- 2 x > - 6$ . Dividing by $- 2$ reverses the symbol, so $x < 3$ . The endpoint is excluded because $x = 3$ makes both original sides equal to $1$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Solve $x + 4 < 9$ .
2 marks
Solve $3 x - 2 \geq 10$ .
2 marks
Solve $- 5 x \leq 20$ .
2 marks
Solve $2 (x + 3) > 5 x - 9$ .
2 marks
Solve $- \frac{x}{2} \geq 3$ and describe its number-line representation.
3 marks
A venue charges a fixed 30 dollars plus 8 dollars per guest. The budget is at most 110 dollars. Find the largest possible whole number of guests.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Solve $x + 4 < 9$ .
2 marks
Subtract $4$ .
$x < 5$
Solve $3 x - 2 \geq 10$ .
2 marks
$3 x \geq 12$ ; divide by $3$ .
$x \geq 4$
Solve $- 5 x \leq 20$ .
2 marks
Divide by $- 5$ and reverse the direction.
$x \geq - 4$
Solve $2 (x + 3) > 5 x - 9$ .
2 marks
$2 x + 6 > 5 x - 9$ gives $15 > 3 x$ .
$x < 5$
Solve $- \frac{x}{2} \geq 3$ and describe its number-line representation.
3 marks
Multiply by $- 2$ and reverse the inequality.
$x \leq - 6$ ; filled circle at $- 6$ , shading left.
A venue charges a fixed 30 dollars plus 8 dollars per guest. The budget is at most 110 dollars. Find the largest possible whole number of guests.
3 marks
$30 + 8 n \leq 110$ gives $n \leq 10$ ; guests are nonnegative whole numbers.
10 guests.

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 simultaneous equations and inequalities with graphical, algebraic and practical interpretations.

Official curriculum source ↗

In this lesson

Understand the idea.

An inequality describes a set

Why a negative reverses direction

See the method in action.

Divide by a negative

Subtract five from both sides.

Divide by negative three and reverse the symbol.

Describe the solution set.

Check the boundary and an interior value.

Brackets and both sides

Expand the left side.

Gather the variable terms and constants.

Divide by the positive coefficient.

Interpret the strict endpoint.

Try it yourself.

Check your understanding

Keep practising