x +13/x-16 less than or equal to solve the following inequality
To solve the inequality x + 13/(x - 16) ≤ 0, we can follow these steps:
Step 1: Find the values of x that make the inequality undefined.
The inequality is undefined when the denominator, x - 16, equals zero. So, we need to find the value of x that makes x - 16 = 0.
Solving this equation, we get:
x - 16 = 0
x = 16
So, x = 16 is excluded from the solution set.
Step 2: Determine the sign changes of the expression x + 13/(x - 16).
To find the sign changes, we need to examine the intervals where x + 13/(x - 16) changes its sign. We can do this by considering the critical values in the given expression. The critical values occur when the numerator (x) equals zero or the denominator (x - 16) equals zero. We have already found that x = 16 makes the inequality undefined, so we need to consider x = 0 as a critical value.
Step 3: Create a sign chart.
We can create a sign chart using x = 0 and x = 16 as critical values. We will be looking at three intervals: (-∞, 0), (0, 16), and (16, +∞).
Interval: (-∞, 0)
Choose a test value in this interval, let's say x = -1. Plug this value into the expression x + 13/(x - 16):
-1 + 13/(-1 - 16) = -1 - 13/(-17) = -1 + 13/17 = -4/17
Since -4/17 is negative, the sign in this interval is negative.
Interval: (0, 16)
Choose a test value in this interval, let's say x = 1. Plug this value into the expression x + 13/(x - 16):
1 + 13/(1 - 16) = 1 - 13/(-15) = 1 + 13/15 = 28/15
Since 28/15 is positive, the sign in this interval is positive.
Interval: (16, +∞)
Choose a test value in this interval, let's say x = 17. Plug this value into the expression x + 13/(x - 16):
17 + 13/(17 - 16) = 17 + 13/(1) = 17 + 13 = 30
Since 30 is positive, the sign in this interval is positive.
Step 4: Determine the solution set.
Based on the sign changes in the intervals, we can conclude:
- The expression x + 13/(x - 16) is negative on the interval (-∞, 0).
- The expression x + 13/(x - 16) is positive on the intervals (0, 16) and (16, +∞).
The inequality states that x + 13/(x - 16) ≤ 0, meaning the expression should be less than or equal to zero.
Therefore, the solution set for this inequality is:
x ∈ (-∞, 0] U (16, +∞)