Inequalities & Changing the Subject

Inequalities and formula rearrangement both test whether you understand balance. With equations, both sides are equal; with inequalities, one side is larger or smaller. With formulae, the goal is to make a chosen variable stand alone so it can be calculated directly.

CSEC questions may ask for a solution set, a number-line representation, or a formula rewritten for a different subject. Write each operation clearly on both sides, and remember that multiplying or dividing an inequality by a negative reverses the sign. That one rule is a common source of lost marks.

Solving Inequalities in One Unknown

Use the same methods as equations, BUT flip the sign when multiplying/dividing by negatives!

$ax + b < c$

Example

$2x + 3 < 11$

$2x < 8$ $x < 4$

Solution: All numbers less than 4

x < 4

Example

$-3x + 2 \geq 8$

$-3x \geq 6$

Flip the sign when dividing by negative!

$x \leq -2$

x ≤ -2

Solving Inequalities with Negative Coefficients

Remember

INEQUALITY GOLDEN RULE: When you multiply or divide both sides by a negative number, FLIP the inequality sign!

$5 > 2$ , but $-5 < -2$

Changing the Subject of a Formula

What Does "Subject" Mean?

The subject is the variable that's isolated on one side (usually the left) of the equals sign.

$V = \pi r^2 h \text{ ← } V \text{ is the subject}$

Changing the subject means rearranging so a DIFFERENT variable is isolated.

$r^2 = \frac{V}{\pi h} \text{ ← } r^2 \text{ is now the subject}$

Why Changing the Subject Matters

Formulae take different forms depending on which quantity is unknown:

$A = \pi r^2$ if you know $r$ and want $A$
$r = \sqrt{\frac{A}{\pi}}$ if you know $A$ and want $r$

Both are the SAME formula, just rearranged for different purposes.

Strategy: Treat It Like Solving an Equation

Rearranging is just like solving for a variable in an equation! Use inverse operations to isolate the target variable.

ExampleExample 1: Linear Formula

Make $x$ the subject of: $y = 3x + 2$

Step 1: Subtract 2 from both sides (undo addition) $y - 2 = 3x + 2 - 2$ $y - 2 = 3x$

Step 2: Divide by 3 (undo multiplication) $\frac{y-2}{3} = \frac{3x}{3}$ $x = \frac{y-2}{3}$

Check: If $y = 8$ , then $x = \frac{8-2}{3} = 2$ . Original: $8 = 3(2) + 2 = 8$ ✓

ExampleExample 2: Formula With Multiplication

Make $r$ the subject of: $C = 2\pi r$

This formula relates circumference ( $C$ ) to radius ( $r$ ).

Step 1: Identify what's attached to $r$

$r$ is being multiplied by $2\pi$

Step 2: Divide both sides by $2\pi$ $\frac{C}{2\pi} = \frac{2\pi r}{2\pi}$ $r = \frac{C}{2\pi}$

Meaning: If you know the circumference, divide by $2\pi$ to find the radius.

ExampleExample 3: Formula With Division

Make $h$ the subject of: $A = \frac{1}{2}bh$

This is the area formula for a triangle.

Step 1: Identify what's with $h$

$h$ is being multiplied by $\frac{1}{2}b$

Step 2: Divide both sides by $\frac{1}{2}b$ (or multiply by its reciprocal $\frac{2}{b}$ ) $\frac{A}{\frac{1}{2}b} = h$

Simplify: $\frac{A}{\frac{1}{2}b} = \frac{2A}{b}$

$h = \frac{2A}{b}$

Meaning: If you know area and base, use this to find height.

Check: If $A = 20$ and $b = 4$ , then $h = \frac{2(20)}{4} = 10$ . Original: $20 = \frac{1}{2}(4)(10) = 20$ ✓

ExampleExample 4: Formula With Powers (Needs Square Root)

Make $a$ the subject of: $V = \frac{1}{3}a^2 h$

This is the volume formula for a pyramid.

Step 1: Multiply both sides by 3 (undo $\frac{1}{3}$ ) $3V = a^2 h$

Step 2: Divide both sides by $h$ (undo multiplication by $h$ ) $\frac{3V}{h} = a^2$

Step 3: Take the square root (undo the square) $a = \sqrt{\frac{3V}{h}}$

Or written another way: $a = \sqrt{\frac{3V}{h}} = \frac{\sqrt{3V}}{\sqrt{h}}$

Meaning: If you know volume and height, use this to find the base side $a$ .

Step-by-Step Strategy

Identify the target variable (the one you want to isolate)
Work backwards through operations using inverses
Order matters:
- Undo addition/subtraction FIRST (loose operations)
- Then undo multiplication/division
- Then undo powers/roots
Do the same operation to BOTH sides

Remember

Common mistakes:

Forgetting to apply the operation to ALL terms
Forgetting that when you divide by something, divide the ENTIRE other side
Forgetting that $\sqrt{x^2} = |x|$ (absolute value), but usually we just write $x$

Changing the Subject of a Formula

What Does "Subject" Mean?

The subject is the variable that's isolated on one side (usually the left) of the equals sign.

V = \pi r^2 h \text{ ← } V \text{ is the subject}

Changing the subject means rearranging so a DIFFERENT variable is isolated.

r^2 = \frac{V}{\pi h} \text{ ← } r^2 \text{ is now the subject}

Why Changing the Subject Matters

Formulae take different forms depending on which quantity is unknown:

A = \pi r^2

if you know

r

and want

A

r = \sqrt{\frac{A}{\pi}}

if you know

A

and want

r

Both are the SAME formula, just rearranged for different purposes.

Strategy: Treat It Like Solving an Equation

Rearranging is just like solving for a variable in an equation! Use inverse operations to isolate the target variable.

ExampleExample 1: Linear Formula

Make $x$ the subject of: $y = 3x + 2$

Step 1: Subtract 2 from both sides (undo addition) $y - 2 = 3x + 2 - 2$ $y - 2 = 3x$

Step 2: Divide by 3 (undo multiplication) $\frac{y-2}{3} = \frac{3x}{3}$ $x = \frac{y-2}{3}$

Check: If $y = 8$ , then $x = \frac{8-2}{3} = 2$ . Original: $8 = 3(2) + 2 = 8$ ✓

ExampleExample 2: Formula With Multiplication

Make $r$ the subject of: $C = 2\pi r$

This formula relates circumference ( $C$ ) to radius ( $r$ ).

Step 1: Identify what's attached to $r$

$r$ is being multiplied by $2\pi$

Step 2: Divide both sides by $2\pi$ $\frac{C}{2\pi} = \frac{2\pi r}{2\pi}$ $r = \frac{C}{2\pi}$

Meaning: If you know the circumference, divide by $2\pi$ to find the radius.

ExampleExample 3: Formula With Division

Make $h$ the subject of: $A = \frac{1}{2}bh$

This is the area formula for a triangle.

Step 1: Identify what's with $h$

$h$ is being multiplied by $\frac{1}{2}b$

Step 2: Divide both sides by $\frac{1}{2}b$ (or multiply by its reciprocal $\frac{2}{b}$ ) $\frac{A}{\frac{1}{2}b} = h$

Simplify: $\frac{A}{\frac{1}{2}b} = \frac{2A}{b}$

$h = \frac{2A}{b}$

Meaning: If you know area and base, use this to find height.

Check: If $A = 20$ and $b = 4$ , then $h = \frac{2(20)}{4} = 10$ . Original: $20 = \frac{1}{2}(4)(10) = 20$ ✓

ExampleExample 4: Formula With Powers (Needs Square Root)

Make $a$ the subject of: $V = \frac{1}{3}a^2 h$

This is the volume formula for a pyramid.

Step 1: Multiply both sides by 3 (undo $\frac{1}{3}$ ) $3V = a^2 h$

Step 2: Divide both sides by $h$ (undo multiplication by $h$ ) $\frac{3V}{h} = a^2$

Step 3: Take the square root (undo the square) $a = \sqrt{\frac{3V}{h}}$

Or written another way: $a = \sqrt{\frac{3V}{h}} = \frac{\sqrt{3V}}{\sqrt{h}}$

Meaning: If you know volume and height, use this to find the base side $a$ .

Step-by-Step Strategy

Identify the target variable (the one you want to isolate)

Work backwards through operations using inverses

Order matters:

Undo addition/subtraction FIRST (loose operations)
Then undo multiplication/division
Then undo powers/roots

Do the same operation to BOTH sides

Remember

Common mistakes:

Forgetting to apply the operation to ALL terms
Forgetting that when you divide by something, divide the ENTIRE other side
Forgetting that $\sqrt{x^2} = |x|$ (absolute value), but usually we just write $x$