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

Select the correct alternative from the given choices.
Find the range of all real values of $x$ if $|3 x+5|<5 x-11$.
(A) $(8, \infty)$
(B) $(-\infty,-5 / 3) \cup(8, \infty)$
(C) $(-5 / 3,8)$
(D) $(-5 / 3, \infty)$

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Key Concepts

Absolute Value Inequalities

This concept involves inequalities that include the absolute value function. Solving these inequalities typically requires considering the definition of absolute value, which measures the distance of a number from zero, and often involves rewriting the inequality as two separate cases to capture when the expression inside is positive or negative.

Case Analysis Using Piecewise Definition

When dealing with absolute value expressions in inequalities, it is necessary to break the problem into cases based on when the expression inside the absolute value is non-negative versus when it is negative. This method, known as piecewise analysis, helps in correctly handling the two different conditions and ensures that the solution set accounts for all possibilities.

Linear Inequalities

After rewriting the absolute value inequality as separate linear inequalities, the next step is to solve these simpler inequalities. Linear inequalities are solved using similar techniques as linear equations, including isolating the variable while paying attention to operations that require reversing the inequality sign, such as multiplying or dividing by negative numbers.

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