3-1 Inequalities and Their Graphs Hubarth Algebra

A solution of an inequality is any number that makes the inequality true. For example, the solution of the inequality 𝑥<3 are all numbers less than 3. So 2, 1, 0, -1,… etc. would be solutions. Ex 1 Identifying Solutions by Mental Math Is each number a solution of x 5? > a. –2 No, –2 5 is not true. > b. 10 Yes, 10 5 is true. > 25 5 c. Yes, 5 5 is true. >

Ex 2 Identifying Solutions by Evaluating Is each number a solution of 3 + 2x < 8? a. –2 b. 3 3 + 2x < 8 3 + 2x < 8 3 + 2(–2) < 8 Substitute for x. 3 + 2(3) < 8 3 – 4 < 8 Simplify. 3 + 6 < 8 –1 < 8 Compare. 9 < 8 –2 is a solution. 3 is not a solution.

Graphing and Writing Inequalities in One Variable You can use a graph to indicate all of the solutions of an inequality. Inequality Graph 𝒙<𝟑 The open circle shows that 3 is not a solution. Shade all values to the left of 3 are solution also. 2 3 4 6 The closed circle shows that -2 is a solution. Shade all values to the right of -2 are solutions also 𝑚≥−2 -4 -2 The closed circle shows that -1 is a solution. Shade all values to the left of -1 are solutions also −1≥𝑎 means 𝑎≤−1 -3 -1 1

} } } } ≤ < ≥ ≤ < > > ≥ Closed circles because they are equal to Shaded to the left ≥ ≤ < > } } Open circles because they are not equal Shaded to the right > ≥

Ex 3 Graphing Inequalities a. Graph d < 3. b. Graph –3 ≥ g.

Ex 4 Writing an Inequality from a Graph Write an inequality for each graph. a. x < 2 Numbers less than 2 are graphed. b. x –3 Numbers less than or equal to –3 are graphed. < c. x > –2 Numbers greater than –2 are graphed. d. x > Numbers greater than are graphed. 1 2

Practice 1. Is each number a solution of 𝑥≥−4.1? a. -5 b. -4.1 c. 8 d. 0 No Yes Yes Yes 2. Is each number a solution of 6x – 3 > 10? a. 1 b. 2 c. 3 d. 4 No No Yes Yes 3. Graph each inequality. a. 𝑎<1 b. 2≥𝑝 𝑝≤2 -1 1 3 2 4 4. Write an inequality for the graph. x ≥10 8 10 12