To solve the inequality [tex]\(3x^2 - 4 \leq 6 - 5x\)[/tex], we follow these steps:
1. Rewrite the inequality:
First, we move all terms to one side of the inequality:
[tex]\[
3x^2 - 4 \leq 6 - 5x
\][/tex]
becomes
[tex]\[
3x^2 - 4 - 6 + 5x \leq 0
\][/tex]
Simplifying,
[tex]\[
3x^2 + 5x - 10 \leq 0
\][/tex]
2. Solve the equality [tex]\(3x^2 + 5x - 10 = 0\)[/tex]:
We use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 3\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = -10\)[/tex].
Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = 5^2 - 4(3)(-10) = 25 + 120 = 145
\][/tex]
The roots are:
[tex]\[
x = \frac{-5 \pm \sqrt{145}}{2 \times 3}
\][/tex]
[tex]\[
x = \frac{-5 \pm \sqrt{145}}{6}
\][/tex]
Compute the approximate decimal values:
[tex]\[
x_1 = \frac{-5 + \sqrt{145}}{6} \approx \frac{-5 + 12.04}{6} \approx \frac{7.04}{6} \approx 1.17
\][/tex]
[tex]\[
x_2 = \frac{-5 - \sqrt{145}}{6} \approx \frac{-5 - 12.04}{6} \approx \frac{-17.04}{6} \approx -2.84
\][/tex]
3. Determine the intervals for the inequality:
The roots split the number line into three intervals: [tex]\((-\infty, -2.84)\)[/tex], [tex]\((-2.84, 1.17)\)[/tex], and [tex]\((1.17, \infty)\)[/tex].
We test points in each interval to see where the inequality [tex]\(3x^2 + 5x - 10 \leq 0\)[/tex] holds.
- For [tex]\(x\)[/tex] within [tex]\((-2.84, 1.17)\)[/tex]:
The quadratic [tex]\(3x^2 + 5x - 10 = 0\)[/tex] changes signs at the roots, meaning it is negative or zero within the interior interval between the roots.
4. Concluding the solution:
Thus, the inequality [tex]\(3x^2 + 5x - 10 \leq 0\)[/tex] holds for:
[tex]\[
-2.84 \leq x \leq 1.17
\][/tex]
So the correct answers are:
[tex]\[
-2.84 \leq x \leq 1.17
\][/tex]
This is the interval within which the quadratic expression [tex]\(3x^2 + 5x - 10\)[/tex] is less than or equal to zero.
