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 Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ 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 $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ 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, represented by |x|, gives the non-negative magnitude of a real number. It essentially measures the distance of a number from zero on the number line, regardless of direction. This concept is crucial when forming equations and inequalities where the sign of the input does not affect the outcome.

Solving Absolute Value Inequalities

Solving absolute value inequalities involves isolating the absolute value expression and then considering the two cases that arise from the definition of absolute value. This typically results in a pair of inequalities, one for when the expression inside is positive and one for when it is negative, which together describe the solution set on the number line.

Graphing Functions and Utilities

Graphing functions using a graphing utility involves inputting equations into designated function slots (such as Y1 and Y2) to visually represent curves on a coordinate system. Graphical representations help in understanding the behavior of functions, finding intersections, and verifying solutions to equations or inequalities by visual inspection.

Intersection Points

Intersection points refer to the points where two graphs meet or cross each other. In the context of solving inequalities or equations, identifying these points is important as they often represent the boundaries of the solution set and help in understanding where one function exceeds another.

Interval Notation

Interval notation is a concise way of describing a set of numbers between two endpoints, indicating whether the endpoints are included or not. It is a standard mathematical notation used for expressing solution sets of inequalities in a clear and unambiguous manner.

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