Question

Solve the following inequality and graph the solution:

(x–6)(x–6)<0

Write the solution as a compound inequality and draw the solution.

a) 6b) x<6 or x>6
c) x≤6 and x≥6
d) −[infinity]

Answer 1

The inequality (x–6)(x–6)<0 has no solution because the product of the two factors cannot be negative, indicating no values of x satisfy the inequality. The correct answer is d) ∞, which stands for an empty set, indicating the absence of a solution.

Step-by-step explanation:

To solve the inequality (x–6)(x–6)<0, we need to find the values of x for which the product of the two factors is less than zero. Since the inequality is a product of two identical binomials, the roots are both x = 6. The factors are positive on either side of this point and negative between them since the parabola associated with x² is positive (concave up), and the inequality states less than zero.

When we consider the values slightly less than 6, say 5.9, the product becomes positive. When we consider values slightly more than 6, say 6.1, the product becomes positive as well. Therefore, there are no values for x that would make the expression negative, so the solution to (x–6)(x–6)<0 is no solution, because there are no x values where the product is less than zero. This can also be interpreted as an empty set, which means there is no interval on the number line that satisfies the inequality. Thus, on the graph, there would be no shaded region, and we cannot write a compound inequality for the solution.

The correct answer to the given question is d) ∞, which means there are no values of x that satisfy the inequality.