2. © William James Calhoun, 2001 OBJECTIVES: You will solve linear inequalities involving more than one operation and find the solution set for a linear inequality when replacement values are given for the variables. When we worked on equations, the section after multiplication and division was over solving equations that involved two or more steps to solution. Remember the steps from Chapter 3 - they still apply to inequalities: (1) Distribute (2) Combine like terms (3) Get all variables on same side (4) Add or subtract numbers from both sides (5) Multiply or divide numbers on both sides 7-3: Solving Multi-Step Inequalities

3. © William James Calhoun, 2001 7-3: Solving Multi-Step Inequalities EXAMPLE 1: Solve 5m - 8 > 12. What is the letter? m What is on the same side as m? positive 5 and negative 8 Which is further from m? the 8 Get rid of it by… adding 8 to both sides. +8 5m > 20 Now get rid of the 5 by… dividing both sides by 5. 5 Pause. Did you divide both sides by a negative number? no Do not change the inequality sign. m > 4 {m | m > 4} EXAMPLE 2: Solve -4w + 9  w - 21. What is the letter? w What is on the same side as w? which one? Move one set of w’s by... subtracting a w from both sides. -w -5w + 9  -21 Now get rid of the -5 by… dividing both sides by -5. -5 Pause. Did you divide both sides by a negative number? yes Change the inequality sign. w  6 Now what is on the same side as the w? negative 5 and positive 9 Which is further from w? the 9 Get rid of it by… subtracting nine from both sides. -9 -5w  -30 {w | w  6}

4. © William James Calhoun, 2001 EXAMPLE 3: Solve 5(k + 4) - 2(k + 6)  5(k + 1) - 1. Then graph. -5k Now get rid of the -2 by… dividing both sides by -2. -2 Pause. Did you divide both sides by a negative number? yes Change the inequality sign. k  2 First thing to do is distribute. 5k + 20 - 2k - 12  5k + 5 - 1 Combine like terms on each side. 3k + 8  5k + 4 What is the letter? k What is on the same side as k? which one? Move one set of k’s by... subtracting 5k from both sides. -2k + 8  4 Now what is on the same side as the k? negative 2 and positive 8 Which is further from w? the 8 Get rid of it by… subtracting eight from both sides. -8 -2k  -4 {k | k  2} 01234 7-3: Solving Multi-Step Inequalities

5. © William James Calhoun, 2001 Remember a few terms from Chapter 3 we will be using again: replacement set - a set of numbers to be plugged into the independent variable solution set - the values of the replacement set that make a true statement Now we will use these terms with inequalities. 7-3: Solving Multi-Step Inequalities

6. © William James Calhoun, 2001 EXAMPLE 4: Determine the solution set for 3x + 6 > 12 if the replacement set for x is {-2, -1, 0, 1, 2, 3, 4, 5}. The solution set is all the replacement set values that make the inequality a true statement. METHOD 1 is plugging in each value and checking. -2 3(-2) + 6 > 12 -6 + 6 > 12 0 > 12 False 3(-1) + 6 > 12 -3 + 6 > 12 3 > 12 False 0 3(0) + 6 > 12 0 + 6 > 12 6 > 12 False 1 3(1) + 6 > 12 3 + 6 > 12 9 > 12 False 2 3(2) + 6 > 12 6 + 6 > 12 12 > 12 False 3 3(3) + 6 > 12 9 + 6 > 12 15 > 12 True 4 3(4) + 6 > 12 12 + 6 > 12 18 > 12 True 5 3(5) + 6 > 12 15 + 6 > 12 21 > 12 True The solution set is: {3, 4, 5}. METHOD 2 is quicker and is on the following slide. 7-3: Solving Multi-Step Inequalities

7. © William James Calhoun, 2001 EXAMPLE 4: Determine the solution set for 3x + 6 > 12 if the replacement set for x is {-2, -1, 0, 1, 2, 3, 4, 5}. The solution set is all the replacement set values that make the inequality a true statement. METHOD 2 involves solving the inequality and then finding the values of the replacement set which are true according to the solution. 3x + 6 > 12 What is the letter? x What is on the same side as the x? positive 3 and positive 6 Which is further from the x? the 6 Get rid of it by… subtracting 6 from both sides. -6 3x > 6 Now get rid of the 3 by... dividing both sides by 3. 3 Pause. Did you divide both sides by a negative number? no Do not change the inequality sign. x > 2 Now, which values of the replacement set are greater than 2? 3, 4, and 5 Therefore the solution set is: {3, 4, 5} 7-3: Solving Multi-Step Inequalities

8. © William James Calhoun, 2001 HOMEWORK Page 403 #17 - 37 odd 7-3: Solving Multi-Step Inequalities