Hello, Girlaaaaaaaa!

Withinequalities, we must be careful . . .quadratic

It says: The product of two quantities is positive.

This is true if: [1] both factors are positive, or [2] both factors are negative.

. . And we must consider both cases.

Both positive: .

So must be greater than 5 and greater than 11.

. . Then: . covers both of them.

Both negative: .

So must be less than 5 and less than 11.

. . Then: . covers both of them.

Solution: .

Interval notation: . 11,\,\infty) " alt="(-\infty,\,5)\:\cup\11,\,\infty) " />

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Here's a rather primitive approach to the problem.

Consider the function: . x-5)(x-11)" alt="y \:=\x-5)(x-11)" />

. . When is this function positive?

We have a parabola

. . When is it above the x-axis?

The parabola opens upward and it has x-intercepts 5 and 11.

The graph is below the x-axis between the intercepts

. . so it isabovethe x-axis to the left of 5 and to the right of 11.

See it?