Solve (x+2)/(x-5) >= 0.

Find Critical Points: Find the critical points of (x+2)/(x-5). The critical points are where the numerator equals zero or the denominator equals zero, since these are the points where the expression could change sign. Numerator: x + 2 = 0 implies x = -2. Denominator: x - 5 = 0 implies x = 5. Critical points: -2, 5Determine Intervals: Determine the intervals using the critical points. The critical points divide the number line into three intervals. Intervals: (-∞, -2), (-2, 5), (5, ∞)Test Sign: Test the sign of (x+2)/(x-5) in each interval. Choose a test point from each interval and substitute it into the inequality to determine the sign of the expression in that interval. Interval (-∞, -2): Choose x = -3. Sign of (x+2)/(x-5) when x = -3: (-3+2)/(-3-5) = (-1)/(-8) = + The expression is positive in this interval. Interval (-2, 5): Choose x = 0. Sign of (x+2)/(x-5) when x = 0: (0+2)/(0-5) = 2/(-5) = - The expression is negative in this interval. Interval (5, ∞): Choose x = 6. Sign of (x+2)/(x-5) when x = 6: (6+2)/(6-5) = 8/1 = + The expression is positive in this interval.Write Solution: Write the solution as a compound inequality. Since we are looking for where (x+2)/(x-5) >= 0, we want the intervals where the expression is positive or zero. The expression is positive in (-∞, -2) and (5, ∞). It is zero when x = -2. Therefore, the solution is x ≤ -2 or x ≥ 5.

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Algebra 2

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\frac{x+2}{x-5} \geq 0

Find Critical Points: Find the critical points of $(x+2)/(x-5)$ . The critical points are where the numerator equals zero or the denominator equals zero, since these are the points where the expression could change sign. Numerator: $x + 2 = 0$ implies $x = -2$ . Denominator: $x - 5 = 0$ implies $x = 5$ . Critical points: $-2, 5$
Determine Intervals: Determine the intervals using the critical points. The critical points divide the number line into three intervals. Intervals: $(-\infty, -2)$ , $(-2, 5)$ , $(5, \infty)$
Test Sign: Test the sign of $(x+2)/(x-5)$ in each interval. $\newline$ Choose a test point from each interval and substitute it into the inequality to determine the sign of the expression in that interval. $\newline$ Interval $(-\infty, -2)$ : Choose $x = -3$ . $\newline$ Sign of $(x+2)/(x-5)$ when $x = -3$ : $(-3+2)/(-3-5) = (-1)/(-8) = +$ $\newline$ The expression is positive in this interval. $\newline$ Interval $(-2, 5)$ : Choose $x = 0$ . $\newline$ Sign of $(x+2)/(x-5)$ when $x = 0$ : $(-\infty, -2)$ $0$ $\newline$ The expression is negative in this interval. $\newline$ Interval $(-\infty, -2)$ $1$ : Choose $(-\infty, -2)$ $2$ . $\newline$ Sign of $(x+2)/(x-5)$ when $(-\infty, -2)$ $2$ : $(-\infty, -2)$ $5$ $\newline$ The expression is positive in this interval.
Write Solution: Write the solution as a compound inequality. $\newline$ Since we are looking for where $(x+2)/(x-5) \geq 0$ , we want the intervals where the expression is positive or zero. $\newline$ The expression is positive in $(-\infty, -2)$ and $(5, \infty)$ . It is zero when $x = -2$ . $\newline$ Therefore, the solution is $x \leq -2$ or $x \geq 5$ .