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QuestionSolve the inequality. Answer should be in interval notation: x+8x+2≥−1\frac{x+8}{x+2} \geq-1 Interval notation solution: □\square No solution Question Help: Video 1 Video 2 Written Example 1 Submit Question

Studdy Solution

STEP 1

1. The inequality involves a rational expression x+8x+2\frac{x+8}{x+2}.
2. We need to find the values of xx that satisfy the inequality x+8x+2≥−1\frac{x+8}{x+2} \geq -1.
3. The solution should be expressed in interval notation.

STEP 2

1. Rearrange the inequality to form a single rational expression.
2. Determine the critical points by setting the numerator and denominator equal to zero.
3. Test intervals between the critical points to determine where the inequality holds.
4. Express the solution in interval notation.

STEP 3

First, rearrange the inequality x+8x+2≥−1\frac{x+8}{x+2} \geq -1 by bringing −1-1 to the left-hand side:
x+8x+2+1≥0\frac{x+8}{x+2} + 1 \geq 0
Combine the terms on the left-hand side over a common denominator:
x+8x+2+x+2x+2≥0\frac{x+8}{x+2} + \frac{x+2}{x+2} \geq 0
(x+8)+(x+2)x+2≥0\frac{(x+8) + (x+2)}{x+2} \geq 0
Simplify the numerator:
2x+10x+2≥0\frac{2x+10}{x+2} \geq 0

STEP 4

Determine the critical points by setting the numerator and denominator equal to zero:
Numerator: 2x+10=02x + 10 = 0
2x=−102x = -10 x=−5x = -5
Denominator: x+2=0x + 2 = 0
x=−2x = -2
The critical points are x=−5x = -5 and x=−2x = -2.

STEP 5

Test intervals between the critical points to determine where the inequality holds. The intervals to test are (−∞,−5)(- \infty, -5), (−5,−2)(-5, -2), and (−2,∞)(-2, \infty).
1. Test x=−6x = -6 in (−∞,−5)(- \infty, -5):
2(−6)+10−6+2=−12+10−4=−2−4=12>0\frac{2(-6) + 10}{-6 + 2} = \frac{-12 + 10}{-4} = \frac{-2}{-4} = \frac{1}{2} > 0
2. Test x=−3x = -3 in (−5,−2)(-5, -2):
2(−3)+10−3+2=−6+10−1=4−1=−4<0\frac{2(-3) + 10}{-3 + 2} = \frac{-6 + 10}{-1} = \frac{4}{-1} = -4 < 0
3. Test x=0x = 0 in (−2,∞)(-2, \infty):
2(0)+100+2=102=5>0\frac{2(0) + 10}{0 + 2} = \frac{10}{2} = 5 > 0
The inequality holds in the intervals (−∞,−5)(- \infty, -5) and (−2,∞)(-2, \infty).

STEP 6

Express the solution in interval notation. Since the inequality is ≥\geq, include the point where the numerator is zero, x=−5x = -5:
The solution in interval notation is:
(−∞,−5]∪(−2,∞)(-\infty, -5] \cup (-2, \infty)
The solution in interval notation is:
(−∞,−5]∪(−2,∞)\boxed{(-\infty, -5] \cup (-2, \infty)}

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