Question

Solve each inequality. Round to the nearest hundredth.3x^2 – 4 ≤ 6 – 5x 1) -2.84 ≤ x ≤ 1.172) no solution3) infinite solutions4) x ≤ –2.84 or x ≥ 1.17

Answer 1

-2.84 ≤ x ≤ 1.17

Step-by-step explanation:

The initial inequality is:

3x² - 4 ≤ 6 - 5x

First, we need to pass all the terms to one side of the inequality, so:

3x² - 4 - 6 + 5x ≤ 6 - 5x - 6 + 5x

3x² + 5x - 10 ≤ 0

Now, we need to find the solutions for the 3x² + 5x - 10 = 0 using the quadratic formula, so:

`\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_1=\frac{-5+\sqrt[]{5^2-4(3)(-10)}}{2(3)}=1.17 \\ x_2=\frac{-5+\sqrt[]{5^2-4(3)(-10)}}{2(3)}=-2.84 \end{gathered}`

It means that we can factorize the inequality as:

3x² + 5x - 10 ≤ 0

(x - 1.17)( x + 2.84) ≤ 0

Now, this expression will be less than 0 if the terms (x - 1.17) and (x + 2.84) has opposite signs. Then, we have two cases:

1. (x - 1.17) is positive and (x + 2.84) is negative

2. (x - 1.17) is negative and (x + 2.84) is positive

For each case, we get:

1. x - 1.17 ≥ 0 and x + 2.84 ≤ 0

x ≥ 1.17 and x ≤ -2.84

Since it is not possible, for this case, we have no solution

2. x - 1.17 ≤ 0 and x + 2.84 ≥ 0

x ≤ 1.17 and x ≥ -2.84

This is the interval -2.84 ≤ x ≤ 1.17.

Therefore, the answer is:

-2.84 ≤ x ≤ 1.17