Solve inequalities in one unknown

• < : less than
• > : greater than
• ≤ : less than or equal to
• ≥ : greater than or equal to

When solving inequalities in one unknown, follow these rules:

1. Collect like terms on one side of the inequality.
2. Adding or subtracting a number does NOT change the inequality direction.
3. Multiplying or dividing by a POSITIVE number does NOT change the direction.
4. Multiplying or dividing by a NEGATIVE number CHANGES the inequality direction.

Solve: $x - 3 \leq 7$

Step 1: Add 3 to both sides (adding doesn't change direction)

$x - 3 + 3 \leq 7 + 3$

$x \leq 10$

Answer: $x \leq 10$

Another Example with a Negative Coefficient

Solve: $-2x > 8$

Step 1: Divide both sides by -2 (dividing by a negative CHANGES the direction)

$\frac{-2x}{-2} < \frac{8}{-2}$

$x < -4$

Answer: $x < -4$

A compound inequality combines two inequalities using "and" or "or".

Example with "and"

Solve: $3 \leq x < 7$

This means $x \geq 3$ and $x < 7$. The solution is all numbers from 3 up to (but not including) 7.

Example with "or"

Solve: $x < 2$ or $x \geq 5$

This means the solution is either less than 2, or greater than or equal to 5.

Solving a Double Inequality

Solve: $10 \leq 2x - 4 < 16$

Step 1: Add 4 to all three parts

$10 + 4 \leq 2x - 4 + 4 < 16 + 4$

$14 \leq 2x < 20$

Step 2: Divide all parts by 2

$\frac{14}{2} \leq x < \frac{20}{2}$

$7 \leq x < 10$

Answer: $7 \leq x < 10$

When showing solutions on a number line:

• Use an empty circle (○) when the endpoint is NOT included (< or >)
• Use a solid circle (●) when the endpoint IS included (≤ or ≥)

For example, $x \leq 5$ is shown as a solid circle at 5 with shading to the left. For $x > 3$, use an empty circle at 3 with shading to the right.

In Tanzania, inequalities are used in everyday situations such as budgeting. For example, if a student's pocket money is 10,000 shillings per week and they need to save at least 2,000 shillings while spending the rest on transport and food, they can write an inequality to find the maximum amount they can spend: $spending \leq 10,000 - 2,000$. This helps them plan their weekly expenses wisely.