Modulus inequalities

Overview of Modulus Inequalities

• Modulus inequalities involve an inequality with a modulus function, noted using absolute bars around an algebraic expression, like

  x

  .

• In maths, the modulus of a number refers to its magnitude or size, regardless of its sign. This means that the modulus of both -4 and 4 is simply 4.

• A modulus inequality looks like

  x

  < a,

  x

  > a,

  x

  ≤ a, or

  x

  ≥ a, where ‘a’ is a real number.

Solving Modulus Inequalities

• To solve a modulus inequality, it’s useful to know that if

  x

  < a, then -a < x < a, and if

  x

  > a, then x < -a or x > a.

• It can also be helpful to solve the corresponding equality first, i.e. find ‘x’ for

  x

  = a, then check the regions separated by these points on a number line to determine where the inequality holds.

Worked Example

• For instance, consider the inequality

  x - 2

  < 3. Firstly, find ‘x’ for

  x - 2

  = 3. This gives x = -1 or x = 5.

• Draw a number line and mark -1 and 5 as critical points. Check each interval. For x < -1,

  x - 2

  > 3. For -1 < x < 5,

  x - 2

  < 3. For x > 5,

  x - 2

  > 3.

• Thus, the solution to

  x - 2

  < 3 is -1 < x < 5.

Exploring Variations

• Inequalities can also have a composite modulus function, like

  3x - 2

  . In this case, treat the whole expression within the absolute bars as a single entity while solving.

• The inequality

  3x - 2

  < 5, would split into -5 < 3x - 2 < 5, and then solving this as usual will give the solution.

Further Tips

• Sketching graph of the modulus function can help visualise and solve the inequality better by seeing where the graph crosses the y-axis.
• Practising a variety of modulus inequalities, with different numbers and variables inside the modulus function, will be most beneficial to fully understand the topic.
• It’s vital to get comfortable solving modulus inequalities, as this concept is fundamental and frequently used in further mathematics.