Save \& Exit Certify Lesson: 6.5 Rational Functions and Rationa...

Question 10 of 12, Step 1 of 1
10/15
Correct

Solve the rational inequality. Express your answer in interval notation.
\[
\frac{x-3}{x+5}< \frac{x}{x-5}
\]

Answer

Step 1 ：Find the critical points of the inequality. These are the values of x that make the numerator or denominator of either fraction equal to zero. For the fraction on the left, the critical points are \(x = 3\) (from the numerator) and \(x = -5\) (from the denominator). For the fraction on the right, the critical points are \(x = 0\) (from the numerator) and \(x = 5\) (from the denominator). So, the critical points are \(x = -5, 0, 3,\) and \(5\).

Step 2 ：Test the intervals determined by these critical points. For \(x < -5\), choose \(x = -6\). Substituting this into the inequality, we get \(-9/-1 < -6/-11\), or \(9 < 6/11\). This is false, so the interval \((-∞, -5)\) is not part of the solution.

Step 3 ：For \(-5 < x < 0\), choose \(x = -1\). Substituting this into the inequality, we get \(-4/4 < -1/-6\), or \(-1 < 1/6\). This is true, so the interval \((-5, 0)\) is part of the solution.

Step 4 ：For \(0 < x < 3\), choose \(x = 1\). Substituting this into the inequality, we get \(-2/6 < 1/-4\), or \(-1/3 < -1/4\). This is false, so the interval \((0, 3)\) is not part of the solution.

Step 5 ：For \(3 < x < 5\), choose \(x = 4\). Substituting this into the inequality, we get \(1/9 < 4/-1\), or \(1/9 < -4\). This is false, so the interval \((3, 5)\) is not part of the solution.

Step 6 ：For \(x > 5\), choose \(x = 6\). Substituting this into the inequality, we get \(3/11 < 6/1\), or \(3/11 < 6\). This is true, so the interval \((5, ∞)\) is part of the solution.

Step 7 ：Therefore, the solution to the inequality is \(\boxed{(-5, 0) ∪ (5, ∞)}\).