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Solve the rafional inequality. White (16)/(5+4x)<=6

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Final answer:

To solve the rational inequality 16/(5+4x) ≤ 6, combine like terms, find the critical values, and test the intervals to determine the solution set.

Step-by-step explanation:

To solve the rational inequality 16/(5+4x) ≤ 6, first subtract 6 from both sides to get 16/(5+4x) - 6 ≤ 0. To find common denominators and combine terms, multiply 6 by (5+4x)/(5+4x) which gives 16/(5+4x) - 6(5+4x)/(5+4x) ≤ 0. Simplify the left side to get (16 - 6(5+4x))/(5+4x) ≤ 0 and further to (-14 - 24x)/(5+4x) ≤ 0.

Next, you determine the critical values by setting the numerator and the denominator to zero separately. For the numerator -14 - 24x = 0 gives x = -14/24 or x = -7/12. For the denominator, 5 + 4x = 0 gives x = -5/4.

Plot these critical values on a number line and choose test points from the intervals created to determine where the inequality is satisfied. Your final answer will be in the form of an interval or a union of intervals depending on where the inequality holds true.

User Barry Fruitman
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