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. $$ |x-4|<3 $$

Key Concepts

Interval Notation

Interval notation is a standardized method to express the set of solutions for an inequality. It provides a concise way to describe ranges of values on the number line that satisfy the inequality, using brackets for inclusive boundaries and parentheses for exclusive boundaries.

Points of Intersection

When graphing two functions to solve an inequality, finding their points of intersection is key, as these points typically mark the boundaries of the solution regions. This approach is particularly useful when using graphing utilities, where the exact coordinates where curves meet help define where one function lies below or above the other.

Absolute Value Inequalities

This concept involves understanding inequalities that include an absolute value expression, such as |x - a| < b. It requires knowing that the inequality can be interpreted as a distance on the number line, and solved by setting up a compound inequality a - b < x < a + b. Mastery of this concept is critical for handling problems where the magnitude of a difference is constrained.

Graphical Representation of Inequalities

This involves plotting functions or expressions on a coordinate system to visually determine solution sets to inequalities. In the context of absolute value inequalities, the graph of the absolute value function is used alongside another function (often a constant) to identify regions where one function is less than the other, providing an intuitive view of the solution.

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