Modulus equations, inequalities and cubic graphs

Learning objectives

Solve equations of the form |ax + b| = c, |ax + b| = cx + d and |ax + b| = |cx + d| algebraically by considering both sign cases, and check every candidate root in the original equation

Solve modulus equations and inequalities graphically by sketching each side and reading off the intersections and the regions where one graph lies above the other

Solve inequalities of the form |ax + b| greater than c and |ax + b| less than c, giving the answer as a union of intervals or as a single interval as appropriate

Sketch the graph of a cubic given in factorised form y = k(x − a)(x − b)(x − c), showing the roots, the y-intercept and the behaviour for large positive and negative x, and account for a repeated factor

Sketch the modulus of a factorised cubic by reflecting the negative parts of the curve in the x-axis, and use the sketch to solve a cubic inequality of the form k(x − a)(x − b)(x − c) ≤ d

Solve equations and inequalities involving the modulus of a quadratic, |ax² + bx + c| = d, greater than d or ≤ d, by treating both sign cases and combining the resulting quadratic conditions, and solve |ax + b| ≤ cx + d where x appears on both sides