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SOLVING INEQUALITIES x 5 9 Step 1: Subtract 5 from both sides x 5 5 9 5 Step 2: Make a Clean Sweep x 5 5 9 5 +5 – 5 equals 0, leaving “x” all by itself=) x 14 Since -9 and -5 have the same sign, add their absolute values. 9 5 14 Since the numbers are both negative, the answer is negative. 14 Final Answer SOLVING INEQUALITIES 5 x 6 Step 1: Divide both sides by -5 5 x 6 5 5 Divide by -5 because: -5 and x are involved in multiplying… The opposite of multiplying is dividing Step 2: Make sure to change inequality sign 5x 6 5 5 Change inequality sign whenever MULTIPLYING OR DIVIDING by a NEGATIVE number=) Step 3: Make a Clean Sweep 5x 6 When dividing a 5 5 The -5’s divide out, leaving “x” by itself. negative number by a negative number, the answer is positive. Final Answer1 x 1.2 or x 1 5 SOLVING INEQUALITIES b 7 4 Step 1: Multiply both sides by 7 7 b 7 4 7 Multiply both sides by 7 because: 7 and b are involved in division… The opposite of dividing is multiplying. Step 2: Make a Clean Sweep 7 b 7 4 7 The 7’s divide out, leaving “x” by itself. b 28 A negative times a positive equals a negative so -4 times 7 is -28. Final Answer SOLVING INEQUALITIES w 4 8 Step 1: Multiply both sides by -4 4 w 4 8 4 Multiply both sides by – 4 because: -4 and w are involved in division… The opposite of dividing is multiplying Step 2: Make sure to change inequality sign 4 w 4 8 4 Change inequality sign whenever MULTIPLYING OR DIVIDING by a NEGATIVE number=) Step 3: Make a Clean Sweep 4 w 4 8 4 The -4’s divide out, leaving “x” by itself. A positive times a negative equals a negative. 8 times -4 is -32. b 32 Final Answer SOLVING INEQUALITIES 4c 2 30 Step 1: Subtract 2 from both sides 4c 2 2 30 2 4c 2 2 30 2 +2 -2 =0, leaving “4c” by itself. Perform adding and subtracting first when solving 2-step inequalities=) 30 – 2 = 28 4c 28 Subtract 2 from both sides because: 2 is involved addition and the opposite of adding is subtracting=) Step 2: Divide both sides by 4 Divide by 4 because: 4 and c are 4c 28 4 4 involved in multiplying… The opposite of multiplying is dividing Step 3: Make a Clean Sweep 4c 28 4 4 The 4’s divide out, leaving “c” by itself. c 7 28 divided by 4 equals 7. Final Answer SOLVING INEQUALITIES 3d 9 12 Step 1: Add 9 to both sides 3d 9 9 12 9 3d 9 9 12 9 -9 +9 =0, leaving “-3d” by itself. Perform adding and subtracting first when solving 2-step inequalities=) 12 + 9 = 21 3d 21 Add 9 to both sides because: 9 is involved subtraction and the opposite of subtracting is adding=) Step 2: Divide both sides by -3 3d 21 3 3 Divide by -3 because: -3 and d are involved in multiplying… The opposite of multiplying is dividing Change inequality sign whenever MULTIPLYING OR DIVIDING by a NEGATIVE number=) Step 3: Make a Clean Sweep 3d 21 3 3 A positive divided by a negative equals a negative 21 divided by -3 equals -7. The -3’s divide out, leaving “d” by itself. d 7 Final Answer SOLVING INEQUALITIES 3y 7y 1 31 Step 1: Combine Like Terms 3y 7y 1 31 3y and -7y are like terms. Since 3y and -7y have The answer keeps the sign of the opposite signs, we subtract integer with the larger absolute the coefficients. (3 and -7 value are the coefficients) 4 y 1 31 3 7 4 Add 1 to both sides because: 1 is Step 2: Add 1 to Both Sides involved subtraction and the opposite of subtracting is adding=) 4y 1 31 4y 1 1 31 1 31 + 1 =32 -1 +1 =0, leaving “-4y” by itself. 4y 32 Step 3:Divide by 4 & Make a Clean Sweep 4y 32 4 4 We change inequality sign whenever we MULTIPLY OR DIVIDE by a NEGATIVE number=) A positive divided by a negative equals a negative. 32 divided by -4 equals -8. The -4’s divide out, leaving “y” by itself. y 8 Final Answer SOLVING INEQUALITIES 8a 2a 9 1 Step 1: Combine Like Terms 8a 2a 9 1 9 and 1 are both integers so we add them. 9 + 1 =10 8a and 2a are like terms so we add the coefficients. 8 + 2 =10 10a 10 Step 2: Divide both sides by 10 10a 10 10 10 Divide by 10 because: 10 and a are involved in multiplying… The opposite of multiplying is dividing Step 3: Make a Clean Sweep 10a 10 10 10 The 10’s divide out, leaving “a” by itself. 10 divided by 10 equals 1. a 1 Final Answer SOLVING INEQUALITIES x 2 7 6 Perform adding and subtracting first when solving 2-step inequalities=) Step 1: Subtract 2 from both sides x 2 2 7 2 6 Since -7 and -2 have the same +2 - 2 =0, leaving “x/6” sign, add their absolute values. by itself. 72 9 x 6 9 Since the numbers are both negative, the answer is negative. Step 2: Multiply both sides by 6 6 x 6 9 6 9 Multiply by 6 because: 6 and x are involved in dividing… The opposite of dividing is multiplying Step 3: Make a Clean Sweep 6 x 6 9 6 6 divided by 6 is 1. A negative times a positive equals a negative. -9 times 6 = -54 x 54 1x=x Final Answer SOLVING INEQUALITIES x 4 5 5 Step 1: Add 5 to both sides x 55 55 4 5 + 5 = 10 -5 + 5 =0, leaving “x/-4” by itself. x 4 10 Step 2: Multiply both sides by -4 4 x 4 10 4 Step 3: Make a Clean Sweep 4 x 4 10 4 4 divided by 4 is 1. 1x=x Perform adding and subtracting first when solving 2-step inequalities=) A positive times a negative equals a negative. 10 times -4 = -40 Multiply by -4 because: -4 and x are involved in dividing… The opposite of dividing is multiplying We change inequality sign whenever we MULTIPLY OR DIVIDE by a NEGATIVE number=) x 40 Final Answer SOLVING INEQUALITIES (b 5) 15 Step 1: Distribute the negative sign to each term inside the parenthesis. 1(b 5) 15 Another way 1(b ) (1)(5) 15 to write the negative sign 1b 5 15 is as a -1=) Step 2: Subtract 5 from both sides. 1b 5 5 15 5 +5 - 5 =0, leaving “-1b” by itself. 1b 10 Step 3: Divide both Sides by -1 1b 10 1 1 15 - 5 = 10 We change inequality sign whenever we MULTIPLY OR DIVIDE by a NEGATIVE number=) Step 4: Make a Clean Sweep A positive divided by a 1b 10 negative equals a negative. 1 1 10 divided by -1 equals -10 The -1’s divide out, leaving “b” by itself. b 10 Final Answer