Modulus Function Set Notation

Question

I have this question

A line l has equation y = ax, where a is a constant.

Given that l intersects y = 2 | x + 4 | − 5 at least once,

(c) find the range of possible values of a, writing your answer in set notation.

i found the answers a>2 and a<5/4. Are these answers correct? If not please help me. If they are how do I write in set notation?

Answer 1

When I teach SAT math, I try to get my students to consider "what does that mean?" instead of "what do I do?". What does y = ax mean? It looks like y =mx + 0 if we interpret it as slope/intercept. So, it is a line crossing the origin, and it's slope is the value of a. So, basically it represents any line crossing the origin. The other expression is a little more difficult to read, because of the absolute value bars. If we replace the confusing part with the phrase "just some number" we have y = 2n -5. Much easier. When n=0 y=-5 Therefore when x = -4 in the original, the y is at it's intercept. Since zero is neither positive nor negative, the absolute value does not matter. We can say the graph passes through the point (-4, -5). For all other values, there will be a positive and negative way to the answer. Let's plot y at -3 and solve for the two x values, -3 = 2n -5 2 = 2n 1= n which means 1=|x+4| x = -3 or -5 so the graph diverges to (-3,-3) and (-5,-3) Let's plot y at -1, -1 = 2n -5 4 = 2n 2 = n which means 2 = |x + 4| x = -2 or -6 so the graph contains the points (-2,-1) and (-6, -1) Now, most lines will pass through that graph two times, and only the line passing along the y-axis will pass through the graph at the point of intersection. So, the answer is the set of all real numbers.