Questions: Question 8, 4.7.71 Solve the rational inequality (a) symbolically and (b) graphically. 4/(x+2) ≥ 0 (a) The solution set is (Type your answer in interval notation.)

Solution Steps

To solve the rational inequality \(\frac{4}{x+2} \geq 0\), we need to determine the values of \(x\) for which the expression is non-negative. This involves finding the critical points where the expression is undefined or equal to zero, and testing intervals around these points to determine where the inequality holds.

1. Identify the critical points by setting the denominator equal to zero and solving for \(x\).
2. Determine the sign of the expression in the intervals defined by these critical points.
3. Use the results to write the solution in interval notation.

The inequality given is:

\[ \frac{4}{x+2} \geq 0 \]

To solve this inequality, we first identify the critical points where the expression is undefined or equal to zero. The expression is undefined when the denominator is zero:

\[ x + 2 = 0 \implies x = -2 \]

The expression \(\frac{4}{x+2}\) is never zero because the numerator is a constant non-zero value (4).

Next, we determine the sign of the expression \(\frac{4}{x+2}\) in the intervals defined by the critical point \(x = -2\). The intervals to consider are:

1. \(x < -2\)
2. \(x > -2\)

• For \(x < -2\), choose a test point, e.g., \(x = -3\): \[ \frac{4}{-3+2} = \frac{4}{-1} = -4 \quad (\text{negative}) \]

• For \(x > -2\), choose a test point, e.g., \(x = 0\): \[ \frac{4}{0+2} = \frac{4}{2} = 2 \quad (\text{positive}) \]

The inequality \(\frac{4}{x+2} \geq 0\) is satisfied when the expression is positive or zero. Since the expression is never zero, we only consider where it is positive:

• The expression is positive for \(x > -2\).

Thus, the solution in interval notation is:

\[ (-2, \infty) \]

Final Answer