\( \DeclareMathOperator{cosec}{cosec} \)

Sign In | Starter Of The Day | Tablesmaster | Fun Maths | Maths Map | Topics | More

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

The modulus (or absolute value) of a number represents its distance from zero on the number line, regardless of its sign. When solving equations and inequalities involving modulus, it's essential to consider both the positive and negative cases. This often results in a system of equations or inequalities to solve.

Key Formulae:

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

For any real numbers \( a \) and \( b \):

1) \( |a| = b \) is equivalent to \( a = b \) or \( a = -b \)

2) \( |a| > b \) is equivalent to \( a > b \) or \( a < -b \)

3) \( |a| < b \) is equivalent to \( -b < a < b \)

**Example 1:**

Solve the equation \( |x - 3| = 5 \).

Using the key formulae, this equation can be broken down into two possible equations:

$$ x - 3 = 5 \quad \text{or} \quad x - 3 = -5 $$

Solving each equation, we get:

\( x = 8 \) or \( x = -2 \)

Thus, the solutions to the equation \( |x - 3| = 5 \) are \( x = 8 \) and \( x = -2 \).

**Example 2:**

Solve the inequality \( |3x \arccos(x)| > 1 \) for \(-1 \le x \le 1 \).

Using the key formulae for modulus inequalities, this inequality can be broken down into two possible inequalities:

$$ 3x \arccos(x) > 1 \quad \text{or} \quad 3x \arccos(x) < -1 $$

$$ x \arccos(x) > \frac{1}{3} $$

$$ x \arccos(x) < -\frac{1}{3} $$

Using the GDC graph plotting function the following solution is obtained:

$$ -1 \le x \lt -0.189 \text{ and } 0.254 \lt x \lt 0.937 $$How do you teach this topic? Do you have any tips or suggestions for other teachers? It is always useful to receive feedback and helps make these free resources even more useful for Maths teachers anywhere in the world. Click here to enter your comments.