Question 1194429
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x = numerator
x+5 = denominator


x/(x+5) is the fraction itself
It is stated to be less than 2/5


x/(x+5) < 2/5


Notice that x = -5 leads to a division by zero error.
This is a natural cutting point to divide the x axis number line into two pieces, and we'll test each piece.


Let's plug in something to the left of x = -5, let's say x = -6
So,
x/(x+5) < 2/5
-6/(-6+5) < 2/5
6 < 2/5
6 < 0.4
which is clearly false
You'll find a false statement results when you plug in anything to the left of x = -5
So we ignore x < -5 and x = -5 itself.


Let's try something to the right of x = -5. I'll try x = 0
x/(x+5) < 2/5
0/(0+5) < 2/5
0 < 2/5
0 < 0.4
That is true. This suggests that x > -5 is the solution set.


However, we need to solve for x like so
x/(x+5) < 2/5
5x < 2(x+5)
5x < 2x+10
5x-2x < 10
3x < 10
x < 10/3
The inequality sign stays the same the entire time. It only flips if we were to multiply or divide both sides by a negative number.


The reality is that yes x > -5 must be the case, but we also need x < 10/3 as well
Put together, the solution set is -5 < x < 10/3


A quick way to verify this is to graph these equations
y = x/(x+5)
y = 2/5
Then notice how the hyperbolic curve is below the horizontal line along the interval -5 < x < 10/3


Graph:
<a href = "https://www.desmos.com/calculator/fmus5i4ao0">https://www.desmos.com/calculator/fmus5i4ao0</a>
Desmos is a free graphing app.
You can click on the intersection of the horizontal line and the curve to see the coordinates of said intersection point.
Keep in mind that 10/3 = 3.333 approximately.


Answer: Anything on the interval -5 < x < 10/3
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