To solve the quadratic inequality [tex]\(2x^2 - 11x \leq 6\)[/tex], we need to follow several steps. Here is the detailed, step-by-step solution:
1. Rewrite the inequality in standard form:
Subtract 6 from both sides to get:
[tex]\[
2x^2 - 11x - 6 \leq 0
\][/tex]
2. Factor the quadratic expression:
We look for two numbers that multiply to [tex]\(2 \times -6 = -12\)[/tex] and add up to [tex]\(-11\)[/tex]. These numbers are [tex]\(-12\)[/tex] and [tex]\(1\)[/tex]. Therefore, we can factor the quadratic expression as:
[tex]\[
2x^2 - 12x + x - 6 \leq 0
\][/tex]
Group the terms:
[tex]\[
(2x^2 - 12x) + (x - 6) \leq 0
\][/tex]
Factor by grouping:
[tex]\[
2x(x - 6) + 1(x - 6) \leq 0
\][/tex]
Factor out the common factor:
[tex]\[
(2x + 1)(x - 6) \leq 0
\][/tex]
3. Find the critical points:
Set each factor equal to zero:
[tex]\[
2x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{2}
\][/tex]
[tex]\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\][/tex]
4. Determine the sign of the expression:
We test intervals around the critical points [tex]\(x = -\frac{1}{2}\)[/tex] and [tex]\(x = 6\)[/tex]. The intervals to test are:
[tex]\[
(-\infty, -\frac{1}{2}), \quad \left(-\frac{1}{2}, 6\right), \quad (6, \infty)
\][/tex]
Choose a test point in each interval and determine if the inequality holds:
- For [tex]\(x \in (-\infty, -\frac{1}{2})\)[/tex], pick [tex]\(x = -1\)[/tex]:
[tex]\[
(2(-1) + 1)((-1) - 6) = (-2 + 1)(-7) = -1 \times -7 = 7 \quad (\text{positive})
\][/tex]
- For [tex]\(x \in \left(-\frac{1}{2}, 6\right)\)[/tex], pick [tex]\(x = 0\)[/tex]:
[tex]\[
(2(0) + 1)(0 - 6) = 1 \times -6 = -6 \quad (\text{negative})
\][/tex]
- For [tex]\(x \in (6, \infty)\)[/tex], pick [tex]\(x = 7\)[/tex]:
[tex]\[
(2(7) + 1)(7 - 6) = 14 + 1)(1) = 15 \times 1 = 15 \quad (\text{positive})
\][/tex]
5. Combine the intervals where the inequality is satisfied:
The inequality [tex]\(2x^2 - 11x - 6 \leq 0\)[/tex] is satisfied in the interval where the expression is non-positive. From our sign analysis, the expression is negative between [tex]\(-\frac{1}{2}\)[/tex] and [tex]\(6\)[/tex], inclusive of the endpoints where the expression is zero.
Hence, the solution to the quadratic inequality [tex]\(2x^2 - 11x \leq 6\)[/tex] is:
[tex]\[
\left[-\frac{1}{2}, 6\right]
\][/tex]
So the correct answer out of the given options is:
[tex]\[
\left[-\frac{1}{2}, 6\right]
\][/tex]
