To solve this inequality, we need to determine the conditions under which the expression
1 / (ax - b) is non-negative, meaning it is either positive or equal to zero.
Step 1: Determine when the expression is undefined.
The expression 1 /𝑎𝑥 − 𝑏 is undefined when the denominator equals zero. Therefore, we need to find the value of 𝑥 that makes the denominator zero:
$$ ax - b = 0 \Rightarrow x = \frac{b}{a} $$
This is a point of discontinuity
so, $$ x = \frac{b}{a} $$
must be excluded from the solution set.
Step 2: Solve the inequality for
$$ \frac{1}{ax - b} \ge 0 $$
Positive case: The reciprocal function
$$ \frac{1}{ax - b} $$ is never equal to zero, so there is no solution for equality.
Final Step: Write the solution in interval notation.
Since the function is undefined at
$$ x = \frac{b}{a} $$
and positive for
$$ x \gt \frac{b}{a} $$
the solution set is:
$$ (\frac{b}{a}, ∞) $$
Graph the number line: The point $$ x = \frac{b}{a} $$ is an excluded point, so we put an open circle at that point and shade the region to the right of it to represent the values where the inequality is satisfied.