Question

Use a sign chart , graph and write interval notation {2-x}/{3+x}≥0

Answer 1

1. Identify critical points: ( x = -3 ) and ( x = -2 ).

2. Test intervals and find non-negative regions:
`\( x \in (-\infty, -3) \cup (-2, 2] \)`.

Explanation:

To solve the inequality
`\((2-x)/(3+x) \geq 0\)`, we begin by identifying critical points where the expression is equal to zero or undefined. In this case, the critical points are (x = -3) and (x = -2). These points partition the real number line into three intervals:
`\((-\infty, -3)\)`,
`\((-3, -2)\)`, and
`\((-2, \infty)\)`. We then test each interval by choosing test points within them and evaluating the expression
`\((2-x)/(3+x)\)` to determine whether it is non-negative or not.

For the interval
`\((-\infty, -3)\)`, we might choose (x = -4) as a test point, and for
`\((-3, -2)\)`, (x = -2.5), and for
`\((-2, \infty)\)`, (x = 0). After evaluating, we find that the expression is non-negative in
`\((-3, -2)\)` and
`\((-2, \infty)\)`. Therefore, the solution set to the inequality is
`\(x \in (-\infty, -3) \cup (-2, 2]\)`.

Understanding the behavior of the expression within each interval is crucial, as it allows us to determine the regions where the inequality is satisfied. In this case, the solution set represents the values of (x) that make the given rational expression non-negative.