Rational Inequalities — Quiz

Question 3 • Algebra II

Which graph shows the solution set of the inequality the fraction with numerator x squared plus 10 x plus 16 and denominator x minus 3 is greater than 0 ?

Answer

Image with description A number line from negative 9 to 3. A segment connects an open circle at negative 8 to an open circle at negative 2. An open circle is at 3 with a ray pointing right.

Image with description A horizontal number line is labeled with integers from negative 9 to 3, with the variable 'x' indicated at the right end. A segment connects a closed circle at negative 8 to a closed circle at negative 2. An open circle is at 3 with a ray extending to the right.

Image with description A number line from negative 9 to 3. A segment connects a closed circle at negative 8 to a closed circle at negative 2. An open circle is at 3 with a ray pointing right.

Image with description A horizontal number line is labeled with integers from negative 9 to 3, with the variable 'x' indicated at the right end. An open circle is at negative 8 with a ray pointing left. A segment connects an open circle at 2 to an open circle at 3.

Explanation

Factor the numerator: x² + 10x + 16 = (x + 8)(x + 2), so the critical values are −8, −2, and 3. Checking the signs shows the fraction is positive on (−8, −2) and (3, ∞). All three endpoints are excluded: the fraction equals zero at −8 and −2, and is undefined at 3. Choice A shows those open endpoints and intervals.

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