Solve the equations and inequalities. For each inequality, graph the solution set and express th

step 1 We want to solve in graph parts A through C of the following problem. In part A, we have the equ

Solve the equations and inequalities. For each inequality,...

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always nonnegative and serves as a basis for understanding how far apart two numbers are.

Solving Absolute Value Equations

When faced with an equation of the form |x - c| = d, the definition of absolute value means that the inside expression can be either d or -d. This leads to two simple equations: x - c = d and x - c = -d. Solving both provides the complete set of solutions.

Solving Absolute Value Inequalities

Absolute value inequalities are solved by interpreting the absolute value as a measure of distance. For an inequality like |x - c| < d, the solution is the set of numbers within a distance d of c, which is expressed as the compound inequality c - d < x < c + d. Conversely, for |x - c| > d, the solution consists of values that are more than d units away from c, giving two separate inequalities: x < c - d or x > c + d.

Interval Notation

Interval notation is a compact form to represent a set of numbers, specifying the endpoints of intervals. Square brackets indicate that endpoints are included in the set, while parentheses indicate exclusion. This notation is used to clearly convey the range of solutions for equations and inequalities.

Graphing on a Number Line

Graphing solutions on a number line involves marking the significant boundary points and shading the regions where the inequality is satisfied. Open circles denote values that are not included in the solution, whereas closed circles indicate included endpoints. This visual representation helps to easily identify and interpret the set of solutions.

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