Solve: [tex]\frac{x^2-14 x+40}{x^2-5 x-36} \leq 0[/tex]
[?] < x \leq \quad \text { or } \quad < x \leq 10
Answer (2)
Factor the numerator and denominator: ( x − 9 ) ( x + 4 ) ( x − 4 ) ( x − 10 ) ≤ 0 .
Identify critical points: x = − 4 , 4 , 9 , 10 .
Test intervals to find where the expression is non-positive.
State the solution: ( − 4 < x ≤ 4 ) ∪ ( 9 < x ≤ 10 ) , which means − 4 < x ≤ 4 or 9 < x ≤ 10 .
Explanation
Problem Analysis We are given the inequality x 2 − 5 x − 36 x 2 − 14 x + 40 ≤ 0 . To solve this inequality, we need to find the intervals where the expression is negative or zero.
Factoring First, we factor the numerator and the denominator: Numerator: x 2 − 14 x + 40 = ( x − 4 ) ( x − 10 ) Denominator: x 2 − 5 x − 36 = ( x − 9 ) ( x + 4 )
Rewriting the Inequality So the inequality becomes: ( x − 9 ) ( x + 4 ) ( x − 4 ) ( x − 10 ) ≤ 0
Finding Critical Points and Intervals The critical points are the roots of the numerator and the denominator, which are x = − 4 , 4 , 9 , 10 . These points divide the number line into the following intervals: ( − ∞ , − 4 ) , ( − 4 , 4 ) , ( 4 , 9 ) , ( 9 , 10 ) , ( 10 , ∞ )
Testing Intervals Now we test a value in each interval to determine the sign of the expression:
( − ∞ , − 4 ) : Let x = − 5 . Then 0"> ( − 5 − 9 ) ( − 5 + 4 ) ( − 5 − 4 ) ( − 5 − 10 ) = ( − 14 ) ( − 1 ) ( − 9 ) ( − 15 ) = 14 135 > 0
( − 4 , 4 ) : Let x = 0 . Then ( 0 − 9 ) ( 0 + 4 ) ( 0 − 4 ) ( 0 − 10 ) = ( − 9 ) ( 4 ) ( − 4 ) ( − 10 ) = − 36 40 < 0
( 4 , 9 ) : Let x = 5 . Then 0"> ( 5 − 9 ) ( 5 + 4 ) ( 5 − 4 ) ( 5 − 10 ) = ( − 4 ) ( 9 ) ( 1 ) ( − 5 ) = − 36 − 5 > 0
( 9 , 10 ) : Let x = 9.5 . Then ( 9.5 − 9 ) ( 9.5 + 4 ) ( 9.5 − 4 ) ( 9.5 − 10 ) = ( 0.5 ) ( 13.5 ) ( 5.5 ) ( − 0.5 ) = 6.75 − 2.75 < 0
( 10 , ∞ ) : Let x = 11 . Then 0"> ( 11 − 9 ) ( 11 + 4 ) ( 11 − 4 ) ( 11 − 10 ) = ( 2 ) ( 15 ) ( 7 ) ( 1 ) = 30 7 > 0
Determining the Solution The inequality is satisfied when the expression is less than or equal to 0. This occurs in the intervals ( − 4 , 4 ] and ( 9 , 10 ] . We include 4 and 10 because the expression is equal to 0 at these points. We exclude -4 and 9 because the expression is undefined at these points.
Final Solution Therefore, the solution to the inequality is: ( − 4 < x ≤ 4 ) ∪ ( 9 < x ≤ 10 ) So, the solution is: − 4 < x ≤ 4 or 9 < x ≤ 10
Examples
Understanding inequalities like this is crucial in various fields. For instance, in economics, you might use them to determine price ranges that maximize profit, considering production costs and market demand. Imagine you're selling handmade bracelets. By setting up an inequality, you can find the price range where your profit stays positive, ensuring you're not losing money on each sale. This involves factoring in the cost of materials, labor, and other expenses. Similarly, in engineering, inequalities help define safe operating conditions for machines, ensuring they don't exceed certain stress or temperature limits.
To solve the inequality x 2 − 5 x − 36 x 2 − 14 x + 40 ≤ 0 , we factor and find critical points to test intervals. The solution is − 4 < x ≤ 4 or 9 < x ≤ 10 .
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