Linear Inequalities: Graphing Solutions and Manipulating Formulas
Explore linear inequalities, compound inequalities, and manipulating formulas to solve for variables. Learn how to graph solutions, solve real-world problems, and answer questions about grades, budgeting, and more using linear inequalities.
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Presentation Transcript
1.6: Solve Linear Inequalities I can: Solve a linear and/or compound inequality and graph the solution set. Manipulate a formula to solve for a particular variable.
Key Vocabulary Linear Inequality Compound Inequality Equivalent Inequalities
Linear Inequality Can be written in one of the following forms where a and b are real numbers and a does not equal 0.
Compound Inequality Consists of two simple inequalities joined by and or or
GUIDED PRACTICE for Examples 1 and 2 Graph the inequality. 1. x > 5 The solutions are all real numbers greater than 5. An open dot is used in the graph to indicate 5is not a solution.
GUIDED PRACTICE for Examples 1 and 2 Graph the inequality. 2. x 3 The solutions are all real numbers less than or equal to 3. A closed dot is used in the graph to indicate 2is a solution.
GUIDED PRACTICE for Examples 1 and 2 Graph the inequality. 3. 3 x < 1 The solutions are all real numbers that are greater than or equalt to 3 and less than 1.
GUIDED PRACTICE for Examples 1 and 2 Graph the inequality. 4. x < 1 orx 2 The solutions are all real numbers that are less than 1 or greater than or equal to 2.
Why? Near the end of the semester or school year, I have students ask me what scores they must get on a final in order to earn a particular grade. You can use linear inequalities to answer this question. You budgeted x amount of dollars for movies. You have already spend $50 of the amount. You can use inequalities to determine staying under budget or reaching your budgeted target.
GUIDED PRACTICE for Examples 3 and 4 Solve the inequality. Then graph the solution. 5. 4x + 9 < 25 4x + 9 < 25 Write original inequality. Subtract 9from each side. 4x < 16 x < 4 Divide each side by 4. 6. 1 3x 14 1 3x 14 Write original inequality. 3x 15 Subtract 1from each side. x 5
GUIDED PRACTICE for Examples 3 and 4 Solve the inequality. Then graph the solution. 7. 5x 7 6x 5x 7 6x Write original inequality. Subtract 5xfrom each side. x > 7 8. 3 x > x 9 3 x > x 9 3 2x > 9 2x > 12 x <6 Write original inequality. Subtract xfrom each side. Subtract 3from each side. Divide each side by 2 and reverse.
EXAMPLE 6 Solve an or compound inequality Solve 3x + 5 11 or 5x 7 23 . Then graph the solution. SOLUTION A solution of this compound inequality is a solution of either of its parts. First Inequality 3x + 5 113x 6x 2 Second Inequality 5x 7 235x 30x 6 Divide each side by 5. Write first inequality. Write second inequality. Add 7 to each side. Subtract 5 from each side. Divide each side by 3.
GUIDED PRACTICE for Examples 5,6, and 7 Solve the inequality. Then graph the solution. 9. 1 < 2x + 7 < 19 1 < 2x + 7 < 19 1 7 < 2x + 7 7 < 19 7 8 < 2x < 12 Write original inequality. Subtract 7 from each expression. Simplify. 4 < x < 6 Divide each expression by 2. ANSWER The solutions are all real numbers greater than 4 and less than 6.
GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 10. 8 x 5 6 8 x 5 Write original inequality. 6 8+5 x 5 + 5 6 + 5 Add 5 to each expression. 3 x 11 11 x Simplify. 3 ANSWER The solutions are all real numbers greater than and equal to 11 and less than and equal to 3.
GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 11. x + 4 9 orx 3 7 SOLUTION A solution of this compound inequality is a solution of either of its parts. First Inequality x + 4 9 Second Inequality x 3 7 Write first inequality. Write second inequality. Add 3 to each side. x 10 x 5 Subtract 4 from each side.
GUIDED PRACTICE for Examples 5,6 and 7 ANSWER The graph is shown below. The solutions are all real numbers. less than or equal to 5 or greater than or equal to 10.
GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 12. 3x 1 < 1 or 2x + 5 11 SOLUTION A solution of this compound inequality is a solution of either of its parts. First Inequality 3x 1< 1 Second Inequality 2x + 5 112x 6 x 3 Divide each side by 5. Write first inequality. Write second inequality. Subtract 5 from each side 3x 0x 0 Add 1 each side . Divide each side by 3.
GUIDED PRACTICE for Examples 5,6 and 7 ANSWER The graph is shown below. The solutions are all real numbers. less than 0 or greater than or equal to 3.
