algebra involving modulus

[half hearth]

I just have no clue with what to do or try.

[forest harness]

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[lean jungle]

The easiest approach would probably be to just draw it.

[half hearth]

A drawing is provided earlier and the minimum/vertex is (2,5). But if that's a method, I don't know what to do. I can attach the pic.

[lean jungle]

Well, you should probably draw it on something that has a grid.

Easier to accurately draw lines that way.

Anyway, you also have y = kx + 4. What do these lines look like for different values of k?

[half hearth]

is there a more algebraic way to do it? i don't want to just guess random numbers and find out the range of values of k that way.

[lean jungle]

Sure. Just solve kx + 4 = 3|x - 2| + 5.

[half hearth]

For one I got x = -5/(k-3) and for the other x = 7/k+3). What do I do from here?

[lean jungle]

I assume the first one is for x ≥ 2 and the second one is for x < 2.
Now, you need both solutions to actually belong to their respective intervals. So, solve a system of inequalities:
-5/(k - 3) ≥ 2
7/(k + 3) < 2

[lean jungle]

So, any progress?

[thin forge]

@half hearth

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