Linear and Absolute Value Inequalities: Interval Notation and Solution Methods

Section P.9 Linear Inequalities and Absolute Value Inequalities

Objectives

This section introduces fundamental concepts and techniques for solving linear and absolute value inequalities, with a focus on interval notation and graphical representation.

• Use interval notation to describe solution sets.

• Find intersections and unions of intervals.

• Solve linear inequalities in one variable.

• Solve compound inequalities.

• Solve absolute value inequalities.

Solving an Inequality

Definition and Solution Set

Solving an inequality involves finding all values that make the inequality true. These values are called solutions, and the complete set of solutions is known as the solution set. A solution is any value that satisfies the inequality.

Interval Notation

Types of Intervals

Interval notation is a concise way to describe sets of real numbers that satisfy certain conditions.

• Open interval $(a, b)$: The set of real numbers between, but not including, $a$ and $b$. $(a, b) = \{ x \mid a < x < b \}$

• Closed interval $[a, b]$: The set of real numbers between, and including, $a$ and $b$. $[a, b] = \{ x \mid a \leq x \leq b \}$

• Infinite interval $(a, \infty)$: The set of real numbers greater than $a$. $(a, \infty) = \{ x \mid x > a \}$

• Infinite interval $(-\infty, b]$: The set of real numbers less than or equal to $b$. $(-\infty, b] = \{ x \mid x \leq b \}$

Parentheses and Brackets

• Parentheses $( )$ indicate endpoints that are not included in the interval.

• Brackets $[ ]$ indicate endpoints that are included in the interval.

• Parentheses are always used with $-\infty$ and $\infty$.

Example 1: Using Interval Notation

Set-Builder Notation and Graphs

• $( -2, 5 )$ $\{ x \mid -2 < x < 5 \}$

• $[1, 3.5]$ $\{ x \mid 1 \leq x \leq 3.5 \}$

• $(-\infty, -1)$ $\{ x \mid x < -1 \}$

Each interval can be represented on a number line, with open circles for excluded endpoints and closed circles for included endpoints.

Intersections and Unions of Intervals

Definitions and Methods

When working with multiple intervals, it is important to understand how to find their intersection and union:

• Intersection: The set of numbers that are in both intervals.

• Union: The set of numbers that are in either interval.

Example 2a: Finding Intersections

Find the intersection of $[1, 3]$ and $(2, 6)$:

• Graph $[1, 3]$ and $(2, 6)$ on a number line.

• The intersection is $(2, 3]$.

Solving Linear Inequalities in One Variable

General Form and Solution Method

A linear inequality in $x$ can be written as:

• $ax + b < 0$

• $ax + b \leq 0$

• $ax + b > 0$

• $ax + b \geq 0$

When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed.

Example: Solving a Linear Inequality

Solve and graph the solution set for $-2 < 5 - 2x$:

• Isolate $x$:

• $-2 < 5 - 2x$

• $-2 - 5 < -2x$

• $-7 < -2x$

• Divide by $-2$ (reverse inequality):

• $\frac{-7}{-2} > x$

• $x < 3.5$

Interval notation: $(-\infty, 3.5)$

Solving Compound Inequalities

Definition and Solution Method

A compound inequality involves two inequalities joined by "and" or "or". The goal is to isolate the variable in the middle.

• Example: $1 < 2x + 3 < 11$

• Subtract 3: $1 - 3 < 2x < 11 - 3$

• $-2 < 2x < 8$

• Divide by 2: $-1 < x < 4$

Interval notation: $(-1, 4)$

Solving Absolute Value Inequalities

General Rules

• If $|u| < c$, then $-c < u < c$

• If $|u| > c$, then $u < -c$ or $u > c$

Example: Solving an Absolute Value Inequality

Solve $|6 - 3x| > 18$:

• Rewrite as two inequalities: $6 - 3x < -18$ or $6 - 3x > 18$

• Solve each:

• $6 - 3x < -18$ $-3x < -24$ $x > 8$

• $6 - 3x > 18$ $-3x > 12$ $x < -4$

Solution set: $x < -4$ or $x > 8$ Interval notation: $(-\infty, -4) \cup (8, \infty)$

Application: Using Linear Equations to Solve Problems

Example: Toll Plaza Cost Comparison

Suppose you can pay a $5 toll per crossing or buy a $25 decal for a 25% discount. Let$x$ be the number of crossings.

• Plan 1: $5x$

• Plan 2: $25 + 0.75(5x)$

• Set up the inequality: $5x > 25 + 3.75x$

• Solve: $1.25x > 25$ $x > 20$

You need to cross more than 20 times for the decal to be the better deal.

Summary Table: Interval Notation Types

Type	Notation	Set-Builder Form	Includes Endpoints?
Open Interval	(a, b)	$\{ x \mid a < x < b \}$	No
Closed Interval	[a, b]	$\{ x \mid a \leq x \leq b \}$	Yes
Half-Open Interval	(a, b]	$\{ x \mid a < x \leq b \}$	Only b
Infinite Interval	(a, \infty)	$\{ x \mid x > a \}$	No
Infinite Interval	$(-\infty, b]$	$\{ x \mid x \leq b \}$	Only b

Additional info: This guide covers the essential algebraic skills for solving inequalities, interpreting interval notation, and applying these concepts to real-world problems, as required in Precalculus.