Question

question
make a sign chart that represents the expression \\(\frac{(-2x - 3)^2 (-7x + 4)^2}{(4x - 7)^3 (2x - 8)^3} \geq 0\\) and use your sign chart to solve the inequality.
plot x-values for which the expression equals zero or is undefined, then click next.
click to plot points. click points to delete them.
\\(\longleftarrow\\) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 \\(\longrightarrow\\) x
answer attempt 1 out of 5

Explanation:

Step1: Find zeros of numerator

Set numerator \((-2x - 3)^2(-7x + 4)^2 = 0\).
For \((-2x - 3)^2 = 0\), solve \(-2x - 3 = 0\) → \(x = -\frac{3}{2}\).
For \((-7x + 4)^2 = 0\), solve \(-7x + 4 = 0\) → \(x = \frac{4}{7}\).

Step2: Find undefined points (denominator zero)

Set denominator \((4x - 7)^3(2x - 8)^3 = 0\).
For \((4x - 7)^3 = 0\), solve \(4x - 7 = 0\) → \(x = \frac{7}{4}\).
For \((2x - 8)^3 = 0\), solve \(2x - 8 = 0\) → \(x = 4\).

Answer:

The \(x\)-values to plot are \(x = -\frac{3}{2}\), \(x = \frac{4}{7}\), \(x = \frac{7}{4}\), and \(x = 4\).