Polynomials Inequalities Practice

Question

Show number line test and state answer in interval notation

10) 6 + 5x - 6x²> 0

Answer 1

Solve -6x² + 5x + 6 = 0 to get the points that go on the number line:

-6x² + 5x + 6 = 0

6x² - 5x - 6 = 0

(3x + 2)(2x -3) = 0

3x + 2 = 0 and 2x - 3 = 0

x = -2/3 and x = 3/2

Put these on a number line:

Those 2 points break the number line up into 3 sections. Pick a convenient value of x from each section and try it to see if it works.

For the left side, I'll pick x = -1

So 6 + 5(-1) - 6(-1)² = 6 - 5 - 6 = -5 and -5 is NOT greater than zero so this section of the number line does NOT work.

For the middle section, I'll pick x = 0

So 6 + 5(0) - 6(0)² = 6 and 6 IS greater than zero so this section works.

For the right section, I'll pick x = 2

So 6 + 5(2) - 6(2)² = 6 + 10 - 24 = -8 and -8 is NOT greater than zero so this section does NOT work.

So the solution is -2/3 < x < 3/2 or, in interval notation (-2/3, 3/2)