Algebra 1 HS Mathematics Unit: 03 Lesson: 01 Solving Inequalities Inequalities are solved for x using the same steps as equations. Clear parenthesis using distribution. Combine variable terms. To keep from having to multiply or divide by a negative number, make sure the final variable term has a positive coefficient. Add/subtract to get constant terms isolated. Multiply/divide to isolate the variable. If you multiply or divide by a negative number you must flip the inequality sign. For final numeric solutions to match the direction of the graphic solution, the variable should be on the left side of the inequality. “Flip” the inequality if needed. Make sure to keep the correct order of the inequality. If 3 x, then x 3. 3 Solutions can be written verbally, symbolically, and graphically. Justify solutions algebraically by testing a value in the selected interval solution or by using the graphing calculator to analyze tables and graphs. Sample Solutions Verbal Symbolic x is less than two x is fewer than two x 2 x is less than or equal to two x is no more than two x is at most two x 2 x is greater than two x is more than two x 2 x 2 x is greater than or equal to two x is no less than two x is at least two Graphic 2 2 2 2 Symbols used in Special Case examples means all real numbers work, shown graphically as means no solution, no graph since no numbers work ©2012, TESCCC Editable Resource Document (ERD)–Content may be modified (DULA terms and conditions still apply) Algebra 1 HS Mathematics Unit: 03 Lesson: 01 Solving Inequalities Sample Problems Solve the following inequalities, giving both a symbolic and number line solution. Use algebraic methods. Check solutions for reasonableness 1. 2(2 – x) < x – 2 Algebraic Method ©2012, TESCCC Justification by Graphing and Table Editable Resource Document (ERD)–Content may be modified (DULA terms and conditions still apply) Algebra 1 HS Mathematics Unit: 03 Lesson: 01 Solving Inequalities 2. 5x 4 6 Algebraic Method Justification by Graph and Table 3. 6(x + 2) – 4x 8 Algebraic Method Justification by Graph and Table 4. 5 9 x 4 x 18 Window: {x: -10, 10, 1} {y: 0, 80, 10} Algebraic Method ©2012, TESCCC Justification by Graph and Table Editable Resource Document (ERD)–Content may be modified (DULA terms and conditions still apply) Algebra 1 HS Mathematics Unit: 03 Lesson: 01 Solving Inequalities 5. 4 x 7 4 4x x Special Case Algebraic Method Justification by Graph and Table 6. 5x 7 5x 8 Special Case Algebraic Method Justification by Graph and Table What do both special cases have in common? How do you know if the answer is empty set or all real numbers? ©2012, TESCCC Editable Resource Document (ERD)–Content may be modified (DULA terms and conditions still apply) Algebra 1 HS Mathematics Unit: 03 Lesson: 01 Solving Inequalities Class-Fun/Home-Fun Name:____________________________ Block:_____ Date:_________________ Directions: Solve each inequality as indicated, giving the solution in both symbolic and number line form. Show all work and solutions on paper. Use a graph and table to solve. 1. 3(x + 4) – 5(x – 1) 5 Use algebraic properties to solve. (Justify solutions by graphs and tables.) 1 2. x 3 9 5 3. 8 3 x 5 4. 5x 23 5x 11 5. 13 17 x ©2012, TESCCC Editable Resource Document (ERD)–Content may be modified (DULA terms and conditions still apply) Algebra 1 HS Mathematics Unit: 03 Lesson: 01 6. 5(x + 3) – 2x 21 7. 3(3x + 1) – (x – 1) 6(x + 10) 8. 4(2x + 1) < 2( x – 1) + 6(x + 2) Write and solve an inequality that models each situation. 9. Beatriz is in charge of setting up a banquet hall. She has five tables that will seat six people each. If no more than 62 people will attend, how many more tables seating four people each will she need? 10. The student council is sponsoring a concert as a fund raiser. Tickets are $3 for students and $5 for adults.The student council wants to raise at least $1000. If 200 students attend, how many adults must attend? ©2012, TESCCC Editable Resource Document (ERD)–Content may be modified (DULA terms and conditions still apply)