Rational Functions — Test

Question 18 • Algebra II

What is the solution set of x over 4 is less than or equal to 9 over x ?

Answer

Image with description A number line from negative 7 to 7. A segment connects an open circle at negative 6 to an open circle at 0. An open circle is at 6 with a ray pointing right.

Image with description A number line from negative 7 to 7. An open circle is at negative 6 with a ray pointing left. A segment connects an open circle at 0 to an open circle at 6.

Image with description A number line from negative 7 to 7. A closed circle is at negative 6 with a ray pointing left. A segment connects an open circle at 0 to a closed circle at 6.

Image with description A number line from negative 7 to 7. A segment connects a closed circle at negative 6 to an open circle at 0. A closed circle is at 6 with a ray pointing right.

Explanation

First note that x cannot equal 0 because it is in the denominator. Rewrite the inequality as (x² − 36)/(4x) ≤ 0; the critical values are −6, 0, and 6. Checking the intervals shows the expression is nonpositive for x ≤ −6 and for 0 < x ≤ 6. Thus the solution is (−∞, −6] ∪ (0, 6], shown by choice D.

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