Select the correct alternative from the given choices. The...

Select the correct alternative from the given choices.
The range of $x$ for which $2 x^{2}-5 x-8 \leq\left|2 x^{2}+x\right|$ is
(A) $\left[-\frac{4}{3}, \infty\right)$
(B) $\left(-\frac{4}{3},-1\right)$
(C) $[-1, \infty)$
(D) $[-1,2]$

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Key Concepts

Interval Notation

Interval notation is a concise way of representing a set of values that satisfy an inequality. It is used to express solution sets in a standardized form, indicating which endpoints are included or excluded in the solution.

Piecewise Analysis

Piecewise analysis is used when a function or expression is defined by different expressions over different parts of its domain. In the context of absolute value inequalities, this technique is crucial to handle the inequality separately in regions defined by the sign of the expression inside the absolute value.

Quadratic Inequalities

Solving quadratic inequalities involves finding the roots of the quadratic equation and testing intervals to determine where the quadratic expression is positive or negative. This method helps in establishing the range of values that satisfy the given inequality.

Absolute Value Inequalities

When solving inequalities containing absolute values, the absolute value is interpreted as a piecewise function. This means the expression inside the absolute value is considered separately for when it is non-negative and when it is negative, leading to different inequalities that must be solved individually.

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