Figure 2 shows a sketch of the graph with equation y = 2|x + 4| - 5 The vertex of the graph is at the point P, shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 11 - 2020 - Paper 2

First, isolate the absolute value term:

$3x + 40 + 5 = 2|x + 4| \Rightarrow 3x + 45 = 2|x + 4|$

Now we divide into two cases based on the definition of absolute values.

Case 1: For $x + 4 \geq 0 \Rightarrow |x + 4| = x + 4$

Substituting this into the equation:

$3x + 45 = 2(x + 4) \Rightarrow 3x + 45 = 2x + 8 \Rightarrow x = -37$

Case 2: For $x + 4 < 0 \Rightarrow |x + 4| = - (x + 4)$

Substituting this into the equation:

$3x + 45 = 2(-x - 4) \Rightarrow 3x + 45 = -2x - 8 \Rightarrow 5x = -53 \Rightarrow x = -10.6$

Thus, the solutions to the equation are $x = -10.6$ and $x = -37$.

To find the range of a, we consider the intersection of the lines:

Setting the equations equal to one another:

$ax = 2|x + 4| - 5$

This will depend on the value of $a$. The line will intersect the absolute function once if the slope $a$ does not exceed the steepness of the graph defined by:

We find the values of a for one intersection:

1. From the vertex of the graph ($x = -4$), the maximum slope of the absolute function is 2.
2. Solving gives us the intervals needed:
   • For intersection: $a < 2$
   • Also consider the line can cross between negative $y$ and positive segments.

Thus the final answer in set notation: $\{ a : a \leq 2.125 \} \cup \{ a : a > 2 \}$