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Solving Inequalities With Multiple Steps copyright©amberpasillas2010 Inequalities Objective – To solve inequalities involving two steps. Easy way Hard way 2x + 3 > 11 2x + 3 > 11 - Undo Multiplication or Division first - Undo Addition or Subtraction first copyright©amberpasillas2010 1 Inequalities Hard way Easy way -Undo Multiply/Divide 1st -Undo Add/Subtract 1st 2x + 3 > 11 2x + 3 > 11 2 2 -3 -3 2x + 3 > 11 2x > 8 2 2 2 2 2 3 11 x + > x>4 2 2 - 23 - 23 x > 8 2 x > 4 copyright©amberpasillas2010 Inequalities 1) Undo the addition or subtraction operation first, by using inverse operations. 2) Undo the multiplication or division second, by using inverse operations. copyright©amberpasillas2010 2 Inequalities REMEMBER: You must always perform the same operation to both sides of an inequality to keep it balanced. To undo an operation, perform its opposite operation to both sides. + → − − → + × → ÷ ÷ → × copyright©amberpasillas2010 Inequalities 1) 2x − 5 ≤ 19 +5 + 5 2x ≤ 24 2 2 x ≤ 12 ← Solid circle 9 10 11 12 13 copyright©amberpasillas2010 3 Inequalities 2) 8x + 1 > 25 −1 − 1 8x > 24 8 8 x>3 Open circle 3 → 4 copyright©amberpasillas2010 Inequalities 3) 15 + x ≥ 6 3 −15 − 15 ( 3) x3 ≥ −9 ( 3) x ≥ −27 Solid circle → -27 0 copyright©amberpasillas2010 4 Inequalities 4) x − 4 < 3 5 +4 + 4 ( 5 ) x5 < 7 ( 5) x < 35 ← Open circle -35 0 35 copyright©amberpasillas2010 Inequalities What Operation Do You Do? FIRST SECOND 1. 3x + 8 < 24 Subtract 8 Divide by 3 2. 5x - 10 > 19 Add 10 Divide by 5 3. x - 4 > 20 2 Add 4 Multiply by 2 4. -30 < -5x + 5 Subtract 5 Divide by -5 copyright©amberpasillas2010 5 Inequalities Think about what you need to do to isolate the variable. Get rid of addition or subtraction first. Second get rid of multiplication or division. copyright©amberpasillas2010 Inequalities 1) 2m + 5 > 19 −5 − 5 2m > 14 2 2 m>7 0 7 copyright©amberpasillas2010 6 Inequalities 2) 7 − 3x ≤ 19 −7 −7 −3x ≤ 12 −3x ≤ 12 −3 − 3 x ≥ −4 -4 0 copyright©amberpasillas2010 Inequalities 3) 5 − 3x > 26 −5 −5 −3x > 21 −3 − 3 x < −7 -14 -7 0 copyright©amberpasillas2010 7 Inequalities 1) Goal: Get the variable alone. 2) Undo operations with their opposite operation. 3) Always do the same thing to both sides of an inequality. 4) Easiest to undo add/subtract before multiply/divide. copyright©amberpasillas2010 Inequalities THE END Take out your study guide! copyright©amberpasillas2010 8 2 Step Inequality Rules #12 x 2 + 6 > 14 -6 -6 x >8 2 2 ( x2 ) > 2 8 x Subtract 6 from both sides. + 6 and -6 cancel out. Multiply both sides by 2. 2 divided by 2 cancels out > 16 Open circle Get the variable alone. → 16 18 copyright©amberpasillas2010 #13 2 Step Inequalities 3x + 1 < 4 −1 − 1 3x < 3 3 3 x <1 -1 0 1 2x − 5 ≥ 7 +5 + 5 2x ≥ 12 2 2 x≥6 2 5 6 7 8 copyright©amberpasillas2010 9 #14 Graphing Inequalities x is less than or equal to -1 All real numbers greater than 4 x>4 x ≤ −1 3 4 -3 -2 -1 0 5 6 copyright©amberpasillas2010 10