Question

3x² - 5x < - 2
a. the solution is (simplify your answer. type your answer in interval notation. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)
b. the solution set is the empty set.
choose the correct graph below.
a.
b.
c.
d.

Explanation:

Step1: Solve the inequality \(3x^{2}-5x < - 2\)

First, rewrite it as \(3x^{2}-5x + 2<0\).
Factor the quadratic expression: \(3x^{2}-5x + 2=(3x - 2)(x - 1)\).
Set \((3x - 2)(x - 1)=0\), then \(x = 1\) or \(x=\frac{2}{3}\).

Step2: Determine the sign of the quadratic function

The quadratic function \(y = 3x^{2}-5x + 2\) is a parabola opening upwards (since the coefficient of \(x^{2}\) is \(3>0\)).
We test the intervals:

• For \(x<\frac{2}{3}\), let \(x = 0\), then \(y=(3\times0 - 2)(0 - 1)=2>0\).
• For \(\frac{2}{3}
• For \(x > 1\), let \(x = 2\), then \(y=(3\times2 - 2)(2 - 1)=(6 - 2)\times1 = 4>0\).

Answer:

The solution in interval notation is \((\frac{2}{3},1)\).
For the graph, since the solution is \(\frac{2}{3}