think about this...

The function is |2x|+4
all the answer choices have a positive slope.
automatically, we disregard the negative half of the absolute value function and only focus on the positive side of f(x) meaning we will not accept any x value less than 0 as an answer for the intersection, giving us the equation 2x+4



now that we are only dealing with the positive sloped part of the absolute value function, take a look at each of the answer choices.
x-2 will never be greater than 2x+4 if X is positive.
try it.
x-2=2x+4
-6=x, this will be the case for any choices that have either a smaller slope, and/or a smaller intercept.
choice E has a larger slope, so lets give that a try.

3x-2=2x+4
x=6
boom, it works.

Choose E

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ASIDE: Some students may be unfamiliar with the above format.
The most common way to define a line or curve is to write y = some expression involving x (e.g., y = 2x - 5)
Just know that graphing the equation y = 2x - 5, is the same as graphing the function f(x) = 2x - 5

Likewise, graphing the function f(x) = |2x|+ 4 is the SAME as graphing the equation y = |2x|+ 4
--------------------------

Let's first find the slope of one of the arms of the graph.

To do so, let's find 2 points that lie on the graph.

f(0) = |2(0)|+ 4
= |0|+ 4
= 4
So, when x = 0, y = 4
(0, 4) is one point.

f(1) = |2(1)|+ 4
= |2|+ 4
= 6
So, when x = 1, y = 6
(1, 6) is another point.

Apply the slope formula to get: slope = (6 - 4)/(1-0) = 2/1 = 2
So, the slope of the red arm is 2


At this point, we might see that the graphs for answer choices C and D both have slope 2.
We know this because each is written in slope y-intercept form

For example, g(x) = 2x - 2 (aka y = 2x - 2) represents a line with slope 2 and a y-intercept of -2
Likewise, g(x) = 2x + 3 represents a line with slope 2 and a y-intercept of 3

Since both lines have the same slope of the red arm of our graph, they are PARALLEL with the red arm.
This means neither line will ever intersect the graph of f

ELIMINATE C and D


Now notice that the graphs for answer choices A and B both have slope 1.
That is, g(x) = x - 2 (aka y = 1x - 2) represents a line with slope 1 and a y-intercept of -2
And g(x) = x + 3 represents a line with slope 1 and a y-intercept of 3
Since both lines have a slope that's LESS THAN 2, both lines will diverge away from the red arm.
So, neither line will ever intersect the graph of f

ELIMINATE A and B

By the process of elimination, the correct answer is E, but let's check it out for "fun"
The graph for answer choice E (y = 3x - 2) has slope 3.
Since the slope of y = 3x - 2 is steeper than the red arm of the graph, we know that the lines will intersect at some point.


Answer: E

Cheers,
Brent

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