Linear Inequalities (AQA A-Level Mathematics): Revision Notes

1. Symbols:

• $\ >$: Greater than
• $\ <$: Less than
• $\ \geq$: Greater than or equal to
• $\ \leq$: Less than or equal to

2. Solving Linear Inequalities:

• Solve the inequality as if it were an equation, performing the same operations on both sides (addition, subtraction, multiplication, division).
• Important Rule: When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

📑Example: Solve the inequality $\ 3x + 2 \leq 8:$

1. Subtract $2$ from both sides: $\ 3x \leq 6.$
2. Divide by 3: $\ x \leq 2.$
3. On a number line, place a closed circle on $2$ and shade to the left, showing that $\ x$ can be any number less than or equal to $2.$ Linear inequalities help us understand ranges of possible values and are used in many real-life situations, like budgeting, where you want to keep your spending within a certain limit.

📑Example: Solve $6x - 10 > 3$

1. Add 10 to both sides:

• $6x - 10 + 10 > 3 + 10$
• $6x > 13$

2. Divide both sides by $6$:

• $x > \frac{13}{6}$

📑Example: Solve $5 - 2x > 10$

Incorrect Solution:

1. Subtract $5$ from both sides:

• $5 - 2x > 10$
• $-2x > 5$

2. Divide by -$2$ (incorrect handling of inequality sign):

• $x > -2.5$

Correct Solution:

1. Subtract $5$ from both sides:

• $5 - 2x > 10$
• $-2x > 5$

2. Divide by -$2$ (correct handling of inequality sign, flipping the inequality):

• $x < -2.5$

📑Example:

1. $1 < 2$
2. Multiply by -$1$:

• $-1 > -2$

📑Example: Solve $5 \leq 2x + 3 \leq 12$

1. Subtract $3$ from all sides:

• $5 - 3 \leq 2x + 3 - 3 \leq 12 - 3$
• $2 \leq 2x \leq 9$

2. Divide all sides by $2$:

• $\frac{2}{2} \leq \frac{2x}{2} \leq \frac{9}{2}$
• $1 \leq x \leq \frac{9}{2}$