Questions: Solve the following rational inequality for (x). [ fracx+3x+5<0 ] Give your answer in interval notation.

Solution Steps

To solve the rational inequality \(\frac{x+3}{x+5}<0\), we need to determine where the expression changes sign. This involves finding the critical points by setting the numerator and denominator to zero separately. The critical points divide the number line into intervals, and we test each interval to see where the inequality holds true. Finally, we express the solution in interval notation, excluding any points where the expression is undefined.

To solve the inequality \(\frac{x+3}{x+5}<0\), we first identify the critical points where the expression is either zero or undefined. These occur when the numerator or the denominator is zero.

1. Numerator: \(x + 3 = 0\)
   Solving for \(x\), we get: \[ x = -3 \]

2. Denominator: \(x + 5 = 0\)
   Solving for \(x\), we get: \[ x = -5 \]

The critical points divide the number line into intervals. We will test each interval to determine where the inequality holds true.

The intervals are:

• \((- \infty, -5)\)
• \((-5, -3)\)
• \((-3, \infty)\)

We will choose a test point from each interval and substitute it into the inequality \(\frac{x+3}{x+5}<0\).

1. Interval \((- \infty, -5)\):
   Choose \(x = -6\): \[ \frac{-6+3}{-6+5} = \frac{-3}{-1} = 3 > 0 \] The inequality is not satisfied.

2. Interval \((-5, -3)\):
   Choose \(x = -4\): \[ \frac{-4+3}{-4+5} = \frac{-1}{1} = -1 < 0 \] The inequality is satisfied.

3. Interval \((-3, \infty)\):
   Choose \(x = 0\): \[ \frac{0+3}{0+5} = \frac{3}{5} > 0 \] The inequality is not satisfied.

The inequality \(\frac{x+3}{x+5}<0\) is satisfied in the interval \((-5, -3)\).

Final Answer