2.3 Linear Inequalities Understand basic terminology related to inequalities Solve linear inequalities symbolically Solve linear inequalities graphically and numerically Solve compound inequalities

Terminology Related to Inequalities Inequalities result whenever the equals sign in an equation is replaced with any one of the symbols ≤, ≥, <, or >. Examples of inequalities include:

Linear Inequality in One Variable (1 of 2) A linear inequality in one variable is an inequality that can be written in the form ax + b > 0 where a ≠ 0. (The symbol > may be replaced by ≥, < or ≤.) Examples of linear inequalities in one variable:

Linear Inequality in One Variable (2 of 2) Using techniques from algebra, we can transform these inequalities into one of the forms ax + b > 0, ax + b ≥ 0, ax + b < 0, or ax + b ≤ 0. For example, by subtracting x from each side of 7x + 5 ≥ x, we obtain the equivalent inequality 6x + 5 ≥ 0. If an inequality is not a linear inequality, it is called a nonlinear inequality.

Properties of Inequalities (1 of 2) Let a, b, and c be real numbers. 1. a < b and a + c < b + c are equivalent. (The same number may be added to or subtracted from each side of an inequality.) 2. If c > 0, then a < b and ac < bc are equivalent. (Each side of an inequality may be multiplied or divided by the same positive number.)

Properties of Inequalities (2 of 2) 3. If c < 0, then a < b and ac > bc are equivalent. (Each side of an inequality may be multiplied or divided by the same negative number provided the inequality symbol is reversed.) Replacing < with ≤ and > with ≥ results in similar properties.

Review of Interval Notation 3 ≤ x ≤ 5 is written as [3, 5] 3 < x < 5 is written as (3, 5) A bracket [ or ] is used when the endpoint is included A parenthesis ( or ) is used when the endpoint is not included.

Example: Solving linear inequalities symbolically (1 of 3) Solve each inequality. Write the solution set in set- builder and interval notation.

Example: Solving linear inequalities symbolically (2 of 3) Solution a. Property 3, multiply both sides by − 3

Example: Solving linear inequalities symbolically (3 of 3) b. Begin by applying the distributive property.

Example: Solving a linear inequality graphically (1 of 3)

Example: Solving a linear inequality graphically (2 of 3)

Example: Solving a linear inequality graphically (3 of 3) Solution The graphs intersect at the point (2, 3).

x-Intercept Method (1 of 2) If a linear inequality can be written as y1 > 0, where > may be replaced by ≥, ≤, or <, then we can solve this inequality by using the x-intercept method. To apply this method for y1 > 0, graph y1 and find the x-intercept. The solution set includes x-values where the graph of y1 is above the x-axis.

x-Intercept Method (2 of 2)

Example: Solving a linear inequality with test values (1 of 3) Solve 3(6 − x) + 5 − 2x < 0 numerically. Solution Make a table of Y1 = 3(6 − X) + 5 − 2X Boundary lies between x = 4 and x = 5.

Example: Solving a linear inequality with test values (2 of 3) Change increment from 1 to 0.1. Boundary is x = 4.6 Test values of x = 4.7, 4.8, 4.9 indicate when x > 4.6, y1 < 0.

Example: Solving a linear inequality with test values (3 of 3)

Compound Inequalities A compound inequality occurs when two inequalities are connected by the word and or or. When the word and connects two inequalities, the two inequalities can sometimes be written as a three-part inequality. x ≥ 40 and x ≤ 70 can be written 40 ≤ x ≤ 70

Example: Solving a three-part inequality symbolically (1 of 2)

Example: Solving a three-part inequality symbolically (2 of 2) b. Multiply each part by 4

Example: Solving inequalities symbolically (1 of 2) Solve the linear inequality symbolically. Express the solution set using interval notation. Solution The parts of this compound inequality can be solved simultaneously.

Example: Solving inequalities symbolically (2 of 2)