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Mathematics

Inequalities & Changing the Subject

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Amari Cross & Matthew Williams
||4 min read
AlgebraFormulaeInequalities

Solving linear inequalities and rearranging formulae to make a different variable 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<cax + b < c

Example

2x+3<112x + 3 < 11

2x<82x < 8 x<4x < 4

Solution: All numbers less than 4

x < 4
Example

−3x+2≥8-3x + 2 \geq 8

−3x≥6-3x \geq 6

Flip the sign when dividing by negative!

x≤−2x \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>25 > 2, but −5<−2-5 < -2

Part 10: Changing the Subject of a Formula — Rearranging for Different Variables

What Does "Subject" Mean?

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

V=πr2h ← V is the subjectV = \pi r^2 h \text{ ← } V \text{ is the subject}

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

r2=Vπh ← r2 is now the subjectr^2 = \frac{V}{\pi h} \text{ ← } r^2 \text{ is now the subject}

Why Changing the Subject Matters

Formulas come in different forms depending on what you need to find:

  • A=πr2A = \pi r^2 if you know rr and want AA
  • r=Aπr = \sqrt{\frac{A}{\pi}} if you know AA and want rr

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.

Example 1: Linear Formula

Example

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

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

Step 2: Divide by 3 (undo multiplication) y−23=3x3\frac{y-2}{3} = \frac{3x}{3} x=y−23x = \frac{y-2}{3}

Check: If y=8y = 8, then x=8−23=2x = \frac{8-2}{3} = 2. Original: 8=3(2)+2=88 = 3(2) + 2 = 8 ✓

Example 2: Formula With Multiplication

Example

Make rr the subject of: C=2πrC = 2\pi r

This formula relates circumference (CC) to radius (rr).

Step 1: Identify what's attached to rr

  • rr is being multiplied by 2π2\pi

Step 2: Divide both sides by 2π2\pi C2π=2πr2π\frac{C}{2\pi} = \frac{2\pi r}{2\pi} r=C2πr = \frac{C}{2\pi}

Meaning: If you know the circumference, divide by 2π2\pi to find the radius.

Example 3: Formula With Division

Example

Make hh the subject of: A=12bhA = \frac{1}{2}bh

This is the area formula for a triangle.

Step 1: Identify what's with hh

  • hh is being multiplied by 12b\frac{1}{2}b

Step 2: Divide both sides by 12b\frac{1}{2}b (or multiply by its reciprocal 2b\frac{2}{b}) A12b=h\frac{A}{\frac{1}{2}b} = h

Simplify: A12b=2Ab\frac{A}{\frac{1}{2}b} = \frac{2A}{b}

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

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

Check: If A=20A = 20 and b=4b = 4, then h=2(20)4=10h = \frac{2(20)}{4} = 10. Original: 20=12(4)(10)=2020 = \frac{1}{2}(4)(10) = 20 ✓

Example 4: Formula With Powers (Needs Square Root)

Example

Make aa the subject of: V=13a2hV = \frac{1}{3}a^2 h

This is the volume formula for a pyramid.

Step 1: Multiply both sides by 3 (undo 13\frac{1}{3}) 3V=a2h3V = a^2 h

Step 2: Divide both sides by hh (undo multiplication by hh) 3Vh=a2\frac{3V}{h} = a^2

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

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

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

Step-by-Step Strategy

  1. Identify the target variable (the one you want to isolate)
  2. Work backwards through operations using inverses
  3. Order matters:
    • Undo addition/subtraction FIRST (loose operations)
    • Then undo multiplication/division
    • Then undo powers/roots
  4. 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 x2=∣x∣\sqrt{x^2} = |x| (absolute value), but usually we just write xx