Question

solve the inequality \\(\frac{x + 5}{x + 2} > -3\\)

give your answer in interval notation.

Explanation:

🆕 New Concept Discovered: Solving Rational Inequalities
Finding critical values and testing intervals on a number line.

Step 1: Set the inequality to zero

To solve a rational inequality, we must first move all terms to one side so that we are comparing the expression to zero.

$$ \frac{x + 5}{x + 2} > -3 $$

Add \(3\) to both sides:

$$ \frac{x + 5}{x + 2} + 3 > 0 $$

Step 2: Combine into a single fraction

To combine the terms, find a common denominator, which is \(x + 2\):

$$ \frac{x + 5}{x + 2} + \frac{3(x + 2)}{x + 2} > 0 $$

$$ \frac{x + 5 + 3x + 6}{x + 2} > 0 $$

$$ \frac{4x + 11}{x + 2} > 0 $$

Step 3: Find the critical points

The critical points occur where the numerator is zero or the denominator is zero. These points divide the number line into test intervals.

• Numerator equals zero:

$$ 4x + 11 = 0 \implies x = -\frac{11}{4} = -2.75 $$

• Denominator equals zero:

$$ x + 2 = 0 \implies x = -2 $$

Our critical points are \(x = -2.75\) and \(x = -2\).

Step 4: Test the intervals

We divide the number line into three intervals based on our critical points:

1. \((-\infty, -2.75)\)
2. \((-2.75, -2)\)
3. \((-2, \infty)\)

We test a value from each interval in our simplified inequality \(\frac{4x + 11}{x + 2} > 0\):

• For \((-\infty, -2.75)\), test \(x = -3\):

$$ \frac{4(-3) + 11}{-3 + 2} = \frac{-12 + 11}{-1} = \frac{-1}{-1} = 1 > 0 \quad \text{(True)} $$

• For \((-2.75, -2)\), test \(x = -2.5\):

$$ \frac{4(-2.5) + 11}{-2.5 + 2} = \frac{-10 + 11}{-0.5} = \frac{1}{-0.5} = -2 > 0 \quad \text{(False)} $$

• For \((-2, \infty)\), test \(x = 0\):

$$ \frac{4(0) + 11}{0 + 2} = \frac{11}{2} = 5.5 > 0 \quad \text{(True)} $$

Step 5: Write the solution in interval notation

Since the inequality is strictly greater than (\(>\)), we do not include the boundary points where the expression equals zero or is undefined.

The inequality is true on the intervals \((-\infty, -2.75)\) and \((-2, \infty)\).

Converting \(-2.75\) back to a fraction gives \(-\frac{11}{4}\).

Answer:

$$ (-\infty, -11/4) \cup (-2, \infty) $$