Solve the inequality algebraically.
$\frac{x+2}{x-13} \leq 1$
The solution is $\square$
(Simplify your answer. Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)
Answer (1)
Subtract 1 from both sides: x − 13 x + 2 − 1 ≤ 0 .
Combine into a single fraction: x − 13 15 ≤ 0 .
Determine the sign: x − 13 < 0 .
Solve for x : x < 13 . The solution in interval notation is ( − ∞ , 13 ) .
Explanation
Understanding the Inequality We are given the inequality x − 13 x + 2 ≤ 1 . Our goal is to solve for x and express the solution in interval notation.
Subtracting 1 First, we subtract 1 from both sides of the inequality to get: x − 13 x + 2 − 1 ≤ 0
Combining into a Single Fraction Next, we combine the left side into a single fraction: x − 13 x + 2 − ( x − 13 ) ≤ 0 x − 13 x + 2 − x + 13 ≤ 0 x − 13 15 ≤ 0
Analyzing the Sign Now, we analyze the sign of the fraction. The numerator is 15 , which is positive. Therefore, for the fraction to be less than or equal to zero, the denominator must be negative. So we have: x − 13 < 0 Note that x − 13 cannot be equal to zero, because then the fraction would be undefined.
Solving for x Solving the inequality x − 13 < 0 , we get: x < 13
Interval Notation Finally, we express the solution in interval notation. Since x must be less than 13 , the solution is the interval ( − ∞ , 13 ) .
Examples
Understanding inequalities like this one is crucial in various real-world scenarios. For instance, imagine you're managing a budget where your expenses must remain below a certain limit to avoid debt. Solving inequalities helps you determine the range of spending that keeps you within your financial constraints. Similarly, in engineering, inequalities are used to ensure that structures can withstand certain loads or stresses without failing. This problem demonstrates a fundamental skill in ensuring that conditions are met within specified boundaries.
