An inequality is a mathematical statement that shows two expressions are not equal, where one is greater than or less than the other.

Inequality Symbols

For example, "TSh 5,000 is less than TSh 10,000" can be written as $5,000 < 10,000$.

Solving Linear Inequalities

A linear inequality in one unknown is similar to a linear equation, but instead of an equals sign, we use an inequality symbol. To solve it, we follow rules similar to solving equations, but with one important exception.

Key Rules:

1. Adding or subtracting the same number on both sides does not change the direction of the inequality.
2. Multiplying or dividing both sides by a positive number does not change the direction.
3. Multiplying or dividing both sides by a negative number reverses the direction of the inequality.

Worked Example

Solve: $x - 4 \leq 7$

Solution:

$x - 4 + 4 \leq 7 + 4$ $x \leq 11$

The solution is $x \leq 11$. This means any number less than or equal to 11 satisfies the inequality.

Number Line Representation

We can show inequalities on a number line:

• Use an open circle (○) when the endpoint is not included ($<$ or $>$).
• Use a closed circle (●) when the endpoint is included ($\leq$ or $\geq$).

For $x \leq 11$, we draw a closed circle at 11 and shade to the left.

For $x > -3$, we draw an open circle at -3 and shade to the right.

Compound Inequalities

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

Using "and"

$3 \leq x < 8$ means $x \geq 3$ and $x < 8$.

The solution is the overlap of both conditions.

Using "or"

$x < -1$ or $x > 4$ means the value of $x$ can satisfy either condition.

Worked Example

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

Solution:

Add 4 to all parts: $10 + 4 \leq 2x - 4 + 4 < 16 + 4$ $14 \leq 2x < 20$

Divide all parts by 2: $\frac{14}{2} \leq x < \frac{20}{2}$ $7 \leq x < 10$

The solution is $7 \leq x < 10$.

The absolute value of a real number $x$, written $|x|$, is the distance of $x$ from zero on the number line. Distance is always non-negative, so absolute value is always positive or zero.

$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

For example:

• $|7| = 7$ (7 is already positive)
• $|-5| = -(-5) = 5$ (the distance from -5 to 0 is 5)

Solving Absolute Value Equations

When solving $|expression| = a$ where $a > 0$, we consider two cases:

1. The expression equals $a$
2. The expression equals $-a$

Worked Example 1

Solve: $|x| = 8$

Solution:

Either $x = 8$ or $x = -8$

So the solution is $x = \pm 8$.

Worked Example 2

Solve: $|x - 3| = 5$

Solution:

Case 1: $x - 3 = 5 \Rightarrow x = 8$

Case 2: $x - 3 = -5 \Rightarrow x = -2$

So the solution is $x = 8$ or $x = -2$.

Worked Example 3

Solve: $|2x + 1| = 7$

Solution:

Case 1: $2x + 1 = 7$ $2x = 6$ $x = 3$

Case 2: $2x + 1 = -7$ $2x = -8$ $x = -4$

So the solution is $x = 3$ or $x = -4$.

Special Cases

• $|x| = 0$ means $x = 0$ (only zero has absolute value zero)
• $|x| = -3$ has no solution (absolute value cannot be negative)

• Inequalities compare values using $<, >, \leq, \geq$
• When solving inequalities, reverse the sign when multiplying or dividing by a negative number
• Use open circles for $<$ or $>$, closed circles for $\leq$ or $\geq$
• Absolute value $|x|$ is the distance of $x$ from zero
• To solve $|expression| = a$ (where $a > 0$), set up two equations: expression = $a$ and expression = $-a$

In Tanzania, inequalities and absolute values are used in many practical situations. For example, when a trader at Kariakoo Market sets a minimum and maximum price for a bag of maize flour (such as "the price must be at least TSh 25,000 but not more than TSh 30,000"), they are using inequalities ($25,000 \leq p \leq 30,000$). Similarly, a bus driver measuring the temperature difference between the morning and afternoon to ensure the air conditioner works properly might calculate $|morning\ temperature - afternoon\ temperature|$ to check that the temperature variation stays within a safe range.