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]