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6-3B Solving Multi-Step Inequalities Algebra 1 Glencoe McGraw-Hill Linda Stamper Solving Multi–Step Inequalities A multi-step inequality is solved by transforming the inequality more than one time. Undo addition or subtraction before undoing multiplication or division or you may make the problem more complicated to solve. Always remember the basic rule when isolating the variable: Whatever you do to one side of the inequality sign, you must also do to the other side of the inequality sign. To solve inequalities that have variables on both sides of the inequality symbol, first collect the variable terms on one side. Solve 6x 22 3x 31. Write the problem. Collect variable terms on one side. Undo addition and simplify. Undo multiplication and simplify. 6x 22 3x 31 6x 6x 22 3x 31 31 31 9 3x 3 3 3 x x 3 To solve inequalities that have variables on both sides of the inequality symbol, first collect the variable terms on one side. Solve 6x 22 3x 31. Write the problem. Collect variable terms on one side. Undo addition and simplify. Undo multiplication and simplify. 6x 22 3x 31 3x 3x 3x 22 31 22 22 3x / 9 3 < 3 x 3 When solving inequalities that contain grouping symbols, first use the distributive property. Solve 6x 3 3x 3x 2. Write problem. Distribute and combine like terms. Collect variable terms on one side. Isolate the variable using inverse operations. 6x 3 3x 3x 2 6x 18 3x 3x 6 9x 18 3x 6 3x 3x 6x 18 6 18 18 6x 24 6 6 x 4 Example 1 64 12w 6w 12w 12w 64 18w 18 18 32 w 9 32 w 9 Solve. Example 2 6x 19 2x 55 2x 2x 4x 19 55 19 19 4x / 36 4 > 4 x 9 Leave the Use good fractionform – place improper. the It negative must be inout front! lowest terms. Example 3 5x 16 2x 9x 15 7 x 16 9x 15 7x 7x 16 2x 15 15 15 1 2x 2 2 1 x 2 1 x 2 If solving an inequality results in a statement that is always true, the solution is the set of all real numbers. If solving an inequality results in a statement that is never true, the solution is an empty set, written as . Solve 8x 2 3x 4 5x 7 8. Write problem. 8x 2 3x 4 5x 7 8 Distribute and 8x 16 3x 12 5x 35 8 combine like terms. 5x 28 5x 27 Collect variable 5x 5x terms on one side. 28 27 Result is a false statement. Copy in your spiral notebook! Solve 3x 4 x 12 2x 12. Write problem. 3x 4 x 12 2x 12 Distribute and 3x 12 x 12 2x 24 combine like terms. 2x 24 2x 24 Collect variable 2x 2x terms on one side. 24 24 all real numbers Result is a true statement. Copy in your spiral notebook! Solve 3x 4 x 12 2x 12. Write problem. 3x 4 x 12 2x 12 Distribute and 3x 12 x 12 2x 24 combine like terms. 2x 24 2x 24 Collect variable Subtract 24 from 24 24 termsside. on one side. 2x 2x each xx Result is a true statement. all real numbers Be sure to simplify the inequality to determine whether it is a true statement. If the inequality is true, the solution is the set of all real numbers. If the inequality is false, the solution is the empty set, written as . Solve. 1) 32 3b 2 6 9b 2 6 6 9b / 8 9 > 9 8 b 9 2) m 81 9m 7 m m 81 8m 7 7 7 88 / 8m 8 < 8 11 m m 11 Solve. 3) 8x 15 4x 5x 6 12x 15 5x 6 5x 5x 7x 15 6 15 15 7x 21 7 7 x 3 4) 7x 2 3x 2 7 x 2 3x 6 3x 3x 10x 2 6 2 2 10x 8 10 10 4 x 5 Solve. 5) 7x 4 11x 8x 22x 1 7x 28 11x 8x 4x 2 4x 28 4x 2 4x 4x 28 2 Solve. 6) 4 y 3 2y 3y 2 y 12 4 y 3 2y 3y 6 y 12 4 y 3 4 y 18 4y 4y 3 18 all real numbers Write an inequality and then solve. 7) One half of a number increased by one is greater than or equal to six less than one fourth of the number. 1 1 n 1 n 6 2 4 2 1 n 1 n 6 4 4 1 1 n n 4 4 1 n 1 6 4 1 1 4 1 4 n 7 1 4 1 n 28 Write an inequality and then solve. 8) Sixteen more than three fourths of a number is more than one eighth of the number less than 2. 3 1 n 16 2 n 4 8 3 6 1 n 16 2 n 4 8 8 1 1 n n 8 8 7 n 16 2 8 16 16 -2 8 7 n 14 8 7 8 7 n 16 6-A4 Pages 311–312 #26–33, 39–42, 44–47.