For the following exercises, input the left-hand side of the...

For the following exercises, input the left-hand side of the inequality as a Y 1 graph in your graphing utility. Enter Y2= the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs(. Find the points of intersection, recall ( $2^{24}$ CALC 5 sintersection, ist curve, enter, $2^{\text {mad }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation. $$|4 x+1|-3 > 2$$

Key Concepts

Absolute Value Function

The absolute value function is defined as the non-negative value of a number, which creates a V-shaped graph when plotted. Understanding its properties is crucial when solving equations or inequalities involving absolute values, as it often leads to splitting the expression into different cases based on the input's sign.

Inequalities

Inequalities involve finding the set of values for which one expression is greater than or less than another. When working with absolute value inequalities, it is important to determine the critical points where the expressions equal each other, then test intervals to define the region that satisfies the inequality.

Graphical Interpretation

Using a graphing utility to visualize functions and their intersections aids comprehension of where one function lies above or below another. This technique is especially useful for absolute value inequalities, as plotting both sides of the equation can reveal the intervals where the inequality holds true.

Intersection Points

Intersection points occur where two graphs meet. For inequalities, these points often represent the boundaries between the regions that satisfy the inequality and those that do not. Identifying these points helps in accurately characterizing the solution set.

Interval Notation

Interval notation is a concise way to represent ranges of numbers that satisfy a given condition or inequality. It utilizes brackets and parentheses to denote whether endpoints are included or excluded, providing a clear and standardized way of writing solution sets.

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