Equations, Inequalities and Graphs

You will be able to

• Solve equations of the form |ax + b| = c, |ax + b| = cx + d, |ax + b| = |cx + d| and |ax² + bx + c| = d by taking both cases, the inside equal to plus or minus the other side.
• Reject a root of |ax + b| = cx + d that makes the right-hand side negative, stating the reason.
• Solve a modulus inequality such as |x − 2| < |2x + 1| from its critical values and a sketch of both graphs, or by squaring both sides, writing the answer as x < −3 or x > 1/3.

You will be able to

• Recognise an equation that is a quadratic in a function of x, such as x², √x, 2ˣ, eˣ, x^(1/3) or lg x, and make a suitable substitution u = f(x).
• Solve the quadratic in u, then back-substitute to find every value of x, using logarithms or powers where needed and leaving answers exact.
• Reject a value of u that is impossible, such as 3ˣ = −1 or √x = −2, stating the reason.

You will be able to

• Sketch y = k(x − a)(x − b)(x − c), showing where it meets both axes and its end behaviour, with the curve touching the x-axis at a repeated root.
• Sketch y = |f(x)| for a cubic f by reflecting in the x-axis the parts of the curve below it.
• Solve a cubic inequality such as (x + 2)(x − 1)(x − 3) ≤ 0 from a sketch, writing the answer as x ≤ −2 or 1 ≤ x ≤ 3.