y = |ax/(x + b)|

= |(ax + ab - ab)/(x + b)|

= |a(x + b)/(x + b) - ab/(x + b)|

= |a - ab/(x + b)|

This tells you that the graph has a horizontal asymptote y = a and vertical asymptote x = -b

Since b < 0, -b > 0

So your vertical asymptote is on the right portion of the y-axis.

Since 0 < a < 1, your horizontal asymptote is between y = 0 and y = 1

When x = 0, y = a(0) / (0 + b) = 0/b = 0


When 0 < x < -b, ax > 0 but x + b < 0

y < 0 since both the numerator and denominator are negative.

The y-values are negative. You'll need to reflect this portion upwards since we have a modulus curve. Draw this portion until you approach the vertical asymptote x = -b, where the graph curves and the x-coordinate gets closer and closer to -b but never touches it.


When x < 0,

both ax < 0 and x + b < 0

Both the numerator and denominator are negative so y is positive. Draw this portion until you approach the horizontal asymptote y = a, where the graph curves and the y-coordinate gets closer and closer to a but never reaches a.

Solving for the inequality :


|ax| ≥ |x + b|

Modulus means an absolute value so it is always ≥ 0

We can divide both sides by |x + b| with no change in sign since we know it is positive.


|ax|/|x + b| ≥ 1

|ax/(x + b)| ≥ 1

So we draw the line y = 1

You will just need to look at the portion of the graph where the y-coordinates are ≥ 1


When y = 1, |ax/(x + b)| = 1

ax/(x + b) = 1 or ax/(x + b) = -1

ax = x + b or ax = -x - b

ax - x = b or ax + x = -b

x(a - 1) = b or x(a + 1) = -b

x = b/(a - 1) or x = -b/(a + 1)


Note that both x-values are positive (you can examine the numerators and denominators' values to see why.)

But, b/(a - 1) = -b/(1 - a)

And -b(1 - a) > -b/(a + 1), since :

a is positive (given by question 0 < a < 1 )

So 1 - a is definitely smaller than a + 1, and a having the same numerator but a smaller denominator means b/(a - 1) is larger.


So when you mark the points on the graph, b/(a - 1) is on the right and -b/(a + 1) is on the left of x = -b


When x increases from -b/(a + 1) to the asymptote x = -b, the curve is increasing and y ≥ 1

Same for when x decreases from b/(a - 1) to x = -b



So to solve the inequality :


-b < x ≤ b/(a - 1) and -b/(a + 1) ≤ x < -b

Or

-b/(a + 1) ≤ x ≤ b/(a - 1) , x ≠ -b

thank you