MMS9_Prep book_Unit6.4.qxp 7/7/09 10:46 AM Page 246 6.4 Solving Linear Inequalities by Using Addition and Subtraction FOCUS Use addition and subtraction to solve inequalities. Consider the inequality !1 " 3. –3 –2 –1 0 1 2 3 4 5 What happens to an inequality if we add the same number to each side? !1 " 3 Add 2 to each side. Left side: !1 # 2 $ 1 Right side: 3 # 2 $ 5 The resulting inequality is still true: 1 " 5 +2 –3 –2 –1 0 +2 1 2 3 4 5 What happens to an inequality if we subtract the same number from each side? !1 " 3 Subtract 2 from each side. Left side: !1 ! 2 $ !3 Right side: 3 ! 2 $ 1 The resulting inequality is still true: !3 " 1 –2 –3 –2 –2 –1 0 1 2 3 4 5 Property of Inequalities When the same number is added to or subtracted from each side of an inequality, the resulting inequality is still true. The strategy that we used to solve an equation can be used to solve an inequality. Isolate the variable to solve. Equation r ! 6 $ –2 r ! 6 # 6 $ !2 # 6 r$4 There is only 1 solution: r $ 4 246 Inequality r ! 6 " !2 r ! 6 # 6 " !2 # 6 r"4 Any number less than 4 is part of the solution. The solution includes 3, 2, and 1, for example. MMS9_Prep book_Unit6.4.qxp 7/8/09 Example 1 4:14 PM Page 247 Solving an Inequality a) Solve the inequality 6 " x # 4. b) Graph the solution on a number line. Solution a) b) 6"x#4 Add 4 to each side to isolate x. 6!4"x#4!4 10 " x This is the same as x & 10. 9 10 11 12 13 14 Place a shaded circle on 10 because 10 is part of the solution. 15 Check 1. Solve each inequality. Graph the solution on the number line. a) p ! 3 " 4 -3 -3 p ! 3 _____ " 4 _____ p " _________________ 1 _____________________ –3 –2 –1 Example 2 0 . 1 b) #5 $ 2 ! a -5 - 2 > 2 + a - 2 ________________________ -7 > a ________________________ -11 -10 -9 -8 -7 -6 2 Solving an Inequality with Variables on Both Sides a) Solve the inequality 3d ! 2 % 2d # 2. b) Graph the solution on a number line. Solution a) b) 3d ! 2 % 2d # 2 3d ! 2 # 2d % 2d # 2 # 2d d ! 2 % #2 d ! 2 # 2 % #2 # 2 d % #4 Subtract 2d from each side. Subtract 2 from each side. Place an open circle on #4 because #4 is not part of the solution. –9 –8 –7 –6 –5 –4 –3 247 MMS9_Prep book_Unit6.4.qxp 7/7/09 10:46 AM Page 248 Check 1. Solve each inequality. Graph the solution on a number line. a) 3z ! 1 & 2z $ 2 b) 4 $ 4x " 6 $ 5x 3z + 1 -1 < 2z -2 - 1 ___________________________ 4 - 4x -4 > 6 - 5x -4 ___________________________ -4x > 2 -5x ___________________________ -4x + 5x > 2 -5x + 5x ___________________________ x>2 ___________________________ 3z < 2z - 3 ___________________________ 3z - 2z < 2z - 3 - 2z ___________________________ ___________________________ z < -3 –7 –6 –5 –4 . –3 1 –2 2 3 4 5 6 Practice 1. Which operation will you perform to each side of the inequality to isolate the variable? a) a ! 1 " 3 b) 2 # m $ 3 subtract 1 ______________ add 3 ______________ c) x $ 4 % 5 d) 6 " 1 $ z add 4 ______________ subtract 1 ______________ 2. Fill in the missing steps to get to the solution. a) x ! 5 " 10 -5 -5 x ! 5 _____ " 10 _____ x" 5 b) 12 & x $ 4 + 4 & x $ 4 _____ +4 12 _____ 16 & x 3. Solve each inequality. Match each inequality with the graph of its solution, below. a) n $ 4 " $2 b) p ! 6 # $2 n>2 ________________________ ________________________ p < -8 ________________________ ________________________ c) u $ 3 % $4 d) 2 ! y " $2 u > -1 ________________________ ________________________ y > -4 ________________________ ________________________ i) ii) iii) iv) 248 c –2 –1 0 1 2 3 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 a d –12 –11 –10 –9 –8 –7 b MMS9_Prep book_Unit6.4.qxp 7/7/09 10:46 AM Page 249 4. a) Solve each inequality. Graph the solution on a number line. i) y ! 3 " #2 ii) 2 ! b $ 5 y + 3 - 3 < -2 -3 ________________________ y < -5 ________________________ –9 –8 –7 –6 . –5 2+b-2<5-2 ________________________ b<3 ________________________ –4 iii) 4 % n # 2 -1 4 2 3 4 3+3<t-3+3 ________________________ 6<t ________________________ . 6>n ________________________ 3 1 iv) 3 $ t # 3 4+2>n-2+2 ________________________ 2 0 5 6 7 5 6 7 8 9 10 b) Write 3 numbers that are possible solutions for each inequality. i) _____________________ answers may vary answers may vary ii) _____________________ may vary iii) answers _____________________ answers may vary iv) _____________________ c) Write a number that is NOT a solution of each inequality. may vary ii) answers ____________ may vary i)answers ____________ answers may vary may vary iii) answers ____________ iv) ____________ 5. Solve, then graph each inequality. a) 6a ! 2 % 5a ! 1 b) 3 ! 2s & s # 3 6a + 2 - 2 > 5a + 1 - 2 ___________________________ 6a > 5a -1 -5a ___________________________ a > -1 ___________________________ ___________________________ –2 . –1 0 1 2 3 6. a) Solve the equation: 4v # 6 ' 3v # 3 4v - 6 + 6 = 3v - 3 + 6 ___________________________ 4v = 3v + 3 ___________________________ 4v - 3v = 3v + 3 -3v ___________________________ ___________________________ v=3 . Graph the solution. 1 2 3 4 5 3 + 2s -3 > s - 3 - 3 ___________________________ 2s > s - 6 ___________________________ 2s - s > s - 6 - s ___________________________ s > -6 ___________________________ -7 -6 -5 -4 -3 -2 b) Solve the inequality: 4v # 6 & 3v # 3 4v - 6 > 3v - 3 ___________________________ 4v - 6 + 6 > 3v - 3 + 6 ___________________________ 4v > 3v + 3 ___________________________ 4v - 3v > 3v + 3 - 3v ___________________________ v>3 Graph the solution. 6 1 2 3 4 5 6 249