Topic: Linear Equations and Inequalities
Subtopic: Solving Linear Inequalities
Overview
There are two major concepts covered in this lesson. (1) You must be able to represent sets in three different ways: by graphing them on a number line (watch "open" vs. "closed" circles), writing them using an inequality (like x<1), or writing them in interval notation (which is the most commonly used method). (2) You must learn to solve a linear inequality.
Caution: When solving an inequality, watch the direction of the inequality sign. Don't forget that if you multiply or divide both sides by a negative number you must reverse the direction of the inequality!
Objectives
By the end of this topic you should know and be prepared to be tested on:
- 2.4.1 Sketch a solution set on a number line
- 2.4.2 Know the difference, on a number line, between an open and closed circle
- 2.4.3 State a solution set in inequality notation and in the closely related set-builder notation
- 2.4.4 State a solution set in interval notation
- 2.4.5 Know the difference, in interval notation, between the regular parenthesis ( and the square bracket [
- 2.4.6 Write intervals in the order of the number line, i.e. with the smaller number on the left
- 2.4.7 Solve linear inequalities algebraically
- 2.4 8 Understand some basic applications of linear inequalities
Terminology
Define: linear inequality in one variable, solution set, inequality notation, set-builder notation, interval notation, open interval, closed interval
Text Notes
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 2.7
- This section starts by introducing the three ways of expressing the solution to an inequality: graphing on a number line, interval notation, and set-builder notation (which is really an expanded form of inequality notation). Note that rather than writing a solution in inequality notation e.g. x ≤ 1, the text often writes it using set-builder notation { x | x ≤ 1 } which is read "the set of all x such that x is less than or equal to 1". The | bar means "such that". The most common notation used especially in later courses is interval notation e.g. (-∞,1]. The chart on page 174 compares the different notations.
- Starting on page 175 the text covers solving linear inequalities via three properties summarized in a chart on page 176. I recommend that you put the "additional property of inequality" in your own words instead of making this so complicated. For instance, "It is OK to add (or subtract) a number to both sides of an inequality." Similarly, the "multiplication properties of inequality" could be combined as, "It is OK to multiply (or divide) both sides of an inequality by a positive number, but if you multiply (or divide) both sides by a negative number you must reverse the direction of the inequality." Remembering to reverse the direction of the inequality sign whenever you multiply or divide by a negative number is very important!
- pg 181 example 8 shows a "no solution" case for linear inequalities. Example 9 shows an "all solution" case for linear inequalities. Pay attention to these special cases.