Modulus Equations & Inequalities |ax+b|

Modulus equations and inequalities are casework exercises: each modulus hides two possibilities, and the marks go to students who surface both on paper.

Equations: $\lvert ax + b \rvert = k$

For $k > 0$, split explicitly:

$\lvert 3x - 2 \rvert = 7$ $\to$ $3x - 2 = 7$ or $3x - 2 = -7$ $\to$ $x = 3$ or $x = -\frac{5}{3}$

The written “or”-split line is the method mark. For $k = 0$ there’s one solution; for $k < 0$, none, and stating “no solutions, since a modulus cannot be negative” can be the entire answer.

Two moduli: $\lvert f(x) \rvert = \lvert g(x) \rvert$, square both sides

$\lvert 2x - 1 \rvert = \lvert x + 4 \rvert$ $\to$ $(2x - 1)^2 = (x + 4)^2$ $\to$ $4x^2 - 4x + 1 = x^2 + 8x + 16$ $\to$ $3x^2 - 12x - 15 = 0$ $\to$ $x^2 - 4x - 5 = 0$ $\to$ $x = 5, x = -1$

Squaring kills both moduli at once and feeds a standard quadratic. The $\pm$ split works too but doubles the cases; squaring is the safer exam route.

Modulus = expression: solve, then check

$\lvert x - 3 \rvert = 2x$ $\to$ $x - 3 = 2x$ gives $x = -3$; $-(x - 3) = 2x$ gives $x = 1$. Check $x = -3$: $\text{RHS} = -6 < 0$, but a modulus can’t equal a negative, reject, with the reason written. Check $x = 1$: $\lvert -2 \rvert = 2$ ✓ $x = 1$

The rejection sentence is a mark, and unchecked invalid solutions are a headline examiner complaint.

Inequalities: two unpackings

For $k > 0$:

• $\lvert ax + b \rvert < k$ $\to$ one sandwich: $-k < ax + b < k$
• $\lvert ax + b \rvert > k$ $\to$ two regions: $ax + b > k$ or $ax + b < -k$

$\lvert 2x - 3 \rvert \ge 5$ $\to$ $2x - 3 \ge 5$ or $2x - 3 \le -5$ $\to$ $x \ge 4$ or $x \le -1$

Write two-region answers as two inequalities joined by “or”, never chained. If you blank on which unpacking is which, sketch $y = \lvert ax + b \rvert$ against $y = k$ (the graph method); the picture settles it in five seconds. The squaring method also works for inequalities between two moduli: $\lvert x - 1 \rvert < \lvert x - 4 \rvert$ $\to$ square, solve the resulting quadratic inequality.

Common mistakes

• Only the positive case solved, half the solutions, every time
• ”<” and ”>” unpackings swapped
• Modulus-equals-expression answers left unchecked
• Squaring used without collecting to zero before solving
• Final answers chained into impossible inequalities

Full topic context: Equations, Inequalities & Graphs notes · the graphical side: graphs of modulus functions.