Sketch the graphs of \(y = \frac{x}{2} + 1\) and \(y = \left| {x - 2} \right|\) on the following axes.

Solve the equation \(\frac{x}{2} + 1 = \left| {x - 2} \right|\).

straight line graph with correct axis intercepts      A1

modulus graph: V shape in upper half plane      A1

modulus graph having correct vertex and y-intercept      A1

[3 marks]

METHOD 1

attempt to solve \(\frac{x}{2} + 1 = x - 2\)     (M1)

\(x = 6\)      A1

Note: Accept \(x = 6\) using the graph.

attempt to solve (algebraically) \(\frac{x}{2} + 1 = 2 - x\)     M1

\(x = \frac{2}{3}\)     A1

[4 marks]

METHOD 2

\({\left( {\frac{x}{2} + 1} \right)^2} = {\left( {x - 2} \right)^2}\)      M1

\(\frac{{{x^2}}}{4} + x + 1 = {x^2} - 4x + 4\)

\(0 = \frac{{3{x^2}}}{4} - 5x + 3\)

\(3{x^2} - 20x + 12 = 0\)

attempt to factorise (or equivalent)       M1

\(\left( {3x - 2} \right)\left( {x - 6} \right) = 0\)

\(x = \frac{2}{3}\)     A1

\(x = 6\)      A1

[4 marks]