Solve: $\frac{x^2-x-12}{x+5} \geq 0$ [?] < x \leq \square or x \geq \square

Factor the numerator: x 2 − x − 12 = ( x − 4 ) ( x + 3 ) .
Identify critical points: x = − 5 , − 3 , 4 .
Create a sign chart to determine intervals where the expression is non-negative.
Write the solution: − 5 < x ≤ − 3 or x ≥ 4 ​ .

Explanation

Analyze the Inequality We are given the inequality x + 5 x 2 − x − 12 ​ ≥ 0 . To solve this inequality, we first need to factor the numerator and find the critical points.

Factor the Numerator The numerator can be factored as x 2 − x − 12 = ( x − 4 ) ( x + 3 ) . So the inequality becomes x + 5 ( x − 4 ) ( x + 3 ) ​ ≥ 0 .

Find Critical Points The critical points are the values of x that make the numerator or the denominator equal to zero. These are x = 4 , x = − 3 , and x = − 5 .

Create a Sign Chart Now we create a sign chart using the critical points − 5 , − 3 , and 4 . These points divide the number line into four intervals: ( − ∞ , − 5 ) , ( − 5 , − 3 ) , ( − 3 , 4 ) , and ( 4 , ∞ ) . We test a value from each interval in the inequality x + 5 ( x − 4 ) ( x + 3 ) ​ ≥ 0 to determine the sign of the expression in that interval.

For ( − ∞ , − 5 ) , let x = − 6 . Then − 6 + 5 ( − 6 − 4 ) ( − 6 + 3 ) ​ = − 1 ( − 10 ) ( − 3 ) ​ = − 30 < 0 .

For ( − 5 , − 3 ) , let x = − 4 . Then 0"> − 4 + 5 ( − 4 − 4 ) ( − 4 + 3 ) ​ = 1 ( − 8 ) ( − 1 ) ​ = 8 > 0 .

For ( − 3 , 4 ) , let x = 0 . Then 0 + 5 ( 0 − 4 ) ( 0 + 3 ) ​ = 5 ( − 4 ) ( 3 ) ​ = − 5 12 ​ < 0 .

For ( 4 , ∞ ) , let x = 5 . Then 0"> 5 + 5 ( 5 − 4 ) ( 5 + 3 ) ​ = 10 ( 1 ) ( 8 ) ​ = 10 8 ​ > 0 .

Determine Intervals The inequality x + 5 ( x − 4 ) ( x + 3 ) ​ ≥ 0 is satisfied when the expression is greater than or equal to zero. From the sign chart, this occurs in the intervals ( − 5 , − 3 ] and [ 4 , ∞ ) . Note that we include − 3 and 4 because the expression is equal to zero at these points, but we exclude − 5 because the expression is undefined at this point.

Write the Solution Therefore, the solution to the inequality is − 5 < x ≤ − 3 or x ≥ 4 .

Final Answer The solution in the requested format is: − 5 < x ≤ − 3 or x ≥ 4 .


Examples
Understanding inequalities like this is crucial in many real-world scenarios. For instance, in business, you might use inequalities to determine the range of sales needed to make a profit. If your profit margin is represented by a rational function, solving an inequality helps you find the minimum sales volume required to ensure the profit margin stays above a certain level. This type of analysis is also used in engineering to ensure systems operate within safe and efficient parameters, and in economics to model market behavior and predict outcomes.

Answered by GinnyAnswer | 2025-07-03

The solution to the inequality x + 5 x 2 − x − 12 ​ ≥ 0 is − 5 < x ≤ − 3 or x ≥ 4 . This is derived from factoring the numerator and determining intervals where the expression is non-negative. After analyzing the intervals created by the critical points, we conclude the solution encompasses the stated ranges.
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Answered by Anonymous | 2025-07-04