Question

solve the polynomial inequality and graph the solution set on a real number line. express the solution set in interval notation.
5x² + 29x - 6 ≤ 0
which number line below shows the graph of the solution set?
oa. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
ob. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
oc. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
od. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
oe. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
of. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
og. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
oh. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Explanation:

Step1: Solve the quadratic equation \(5x^{2}+29x - 6=0\)

Use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(ax^{2}+bx + c = 0\). Here \(a = 5\), \(b=29\), \(c=-6\).

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We get \(x_1=\frac{-29 + 31}{10}=\frac{2}{10}=\frac{1}{5}\) and \(x_2=\frac{-29-31}{10}=\frac{-60}{10}=-6\)

Step2: Determine the sign of the quadratic function \(y = 5x^{2}+29x - 6\)

The quadratic function \(y = 5x^{2}+29x - 6\) has \(a = 5>0\), so the parabola opens upwards.
The inequality \(5x^{2}+29x - 6\leq0\) holds for values of \(x\) between the roots (including the roots).

Answer:

The solution set in interval notation is \([-6,\frac{1}{5}]\). So the number - line that shows the graph of the solution set is the one with a closed - circle at \(x=-6\) and \(x = \frac{1}{5}\) and the line segment between them.