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 12 - 2020 - Paper 2

To solve the equation, we first rewrite it:

$3x + 40 = 2|x + 4| - 5$

This simplifies to:

$3x + 45 = 2|x + 4|$

Now we will consider two cases based on the definition of the absolute value.

Case 1: When $x + 4 \geq 0$ (i.e., $x \geq -4$):

Then, $|x + 4| = x + 4$ and substituting gives us:

$3x + 45 = 2(x + 4)$

Solving for x: $3x + 45 = 2x + 8$ $x = -37.$

This solution is only valid if $-37 \geq -4$, which it is not.

Case 2: When $x + 4 < 0$ (i.e., $x < -4$):

Then, $|x + 4| = -(x + 4)$ and substituting gives:

$3x + 45 = -2(x + 4)$

$3x + 45 = -2x - 8$ $5x = -53$

Thus, $x = -10.6.$

To determine the values of a such that the line $y = ax$ intersects the graph of $y = 2|x + 4| - 5$ at least once, we need to analyze the intersection.

The intersection occurs if:

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

This leads to two branches based on the absolute value

1. For $x o -4$ (coming from the left):

y approaches $0 - 5 = -5.$

2. As $x$ gets larger, the line will yield positive values, while the expression will continue to increase because of the absolute function.

Thus, to have an intersection, a must be: $a > 2$

In terms of set notation, the answer is: $\{a: a > 2\}$.