Solve x-5/x+2 ≤ x-8/x+6 using an interval table or by sketching. (5 marks)
To solve the inequality \(\frac{x-5}{x+2} \leq \frac{x-8}{x+6}\), we can start by finding the critical values where the inequality may change.
First, we identify the values that make the denominators of the fractions zero. For the left-hand side, \(x = -2\) will make the denominator zero. For the right-hand side, \(x = -6\) will make the denominator zero.
Next, we should consider the values where the numerators are zero. For the left-hand side, \(x = 5\) makes the numerator zero. For the right-hand side, \(x = 8\) makes the numerator zero.
Now let's create an interval table:
Interval Expression Sign
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(-∞, -6) \(\frac{x-5}{x+2} - \frac{x-8}{x+6}\) +
(-6, -2) \(\frac{x-5}{x+2} - \frac{x-8}{x+6}\) -
(-2, 5) \(\frac{x-5}{x+2} - \frac{x-8}{x+6}\) +
(5, ∞) \(\frac{x-5}{x+2} - \frac{x-8}{x+6}\) -
To determine the sign in each interval, choose a test point from within the interval. For example, in the interval (-∞, -6), we can choose x = -7. Substitute this test point into the expression \(\frac{x-5}{x+2} - \frac{x-8}{x+6}\) to evaluate if it is positive or negative. Repeat this process for each interval.
Now, to sketch the graph of the inequality, draw a number line and mark the critical values (-6 and -2) and the points where the numerator is zero (5 and 8). Then, consider the sign of the expression within each interval and indicate it as positive (+) or negative (-). On the number line, the inequality will be satisfied in the intervals where the expression is less than or equal to zero (negative).
Based on the interval table and sketching process, we can conclude that the solution to the inequality \(\frac{x-5}{x+2} \leq \frac{x-8}{x+6}\) is \(x \in (-6, 5] \cup (8, \infty)\).