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Algebra II Recipe: Solving Inequalities with One Variable and Compound Sentences
A. Inequality Symbols
  1. > means "is greater than"
  2. < means "is less than"
  3. ≥ means "is greater than or equal to"
  4. ≤ means "is less than or equal to"
B. Steps for Solving Inequalities with One Variable
  1. Perform the distributive property on each side.
  2. Combine like terms on each side.
  3. Add or subtract to get the variable terms on the same side. (Side of the largest coefficient.)
  4. Add or subtract to move the number term to the opposite side of the variable term.
  5. Multiply or divide to move the coefficient.
    • If you multiply or divide both sides of an inequality by a negative number, the inequality symbol changes directions.
ExamplesExamples:
x - 8 < 15
4y + 3 > 7
13 - 7n ≤ -8
3x ≥ 11x + 4
C. Steps for Graphing the Solutions to Inequalities with One Variable
          ** Make sure the variable is on the LEFT in all solutions.
  1. A solution with >:
    • Graph an open circle on the number. (The number is not part of the solution.)
    • A dark bar with an arrow goes to the right of the circle.
  2. A solution with <:
    • Graph an open circle on the number. (The number is not part of the solution.)
    • A dark bar with an arrow goes to the left of the circle.
  3. A solution with ≥:
    • Graph a solid circle on the number. (The number is part of the solution.)
    • A dark bar with an arrow goes to the right of the circle.
  4. A solution with ≤:
    • Graph a solid circle on the number. (The number is part of the solution.)
    • A dark bar with an arrow goes to the left of the circle.
ExamplesExamples:
2(3x - 2) < 4x + 8
3(4x - 6) ≥ 6(x + 2)
x + 3 ≤ 2(x - 4)
-x + 4 < 2(x - 8)
-2(x + 3) < 4x - 7
D. Solving a Compound Inequality with "AND"
  1. Isolate the variable in the middle.
    • Distribute in the middle if possible.
    • Combine like terms in the middle if possible.
    • Add or subtract the number term on each side of both inequality symbols.
    • Multiply or divide by the coefficient on each side of both inequality symbols.
  2. If the solution contains greater than symbols, rotate the whole solution around to get less than symbols. (This would happen when you multiply or divide by a negative.)
  3. Graph the solution.
    • One of the circles goes on each number in the solution.
    • A darkened bar is graphed between the two circles.
ExamplesExamples:
-4 < x + 2 ≤ 4
-3 ≤ 2x + 1 ≤ 5
17 < 5 - 3x < 29
E. Solving a Compound Inequality with "OR"
          ** It's written like 8 + 2x < 6 OR 3x - 2 > 13
  1. Solve each inequality.
  2. The solution must be written with two inequalities connected with "OR".
  3. Graph each inequality.
    • One of the circles goes on each number in the solution.
    • The dark bar with an arrow is graphed in the direction indicated by the symbol with the number.
    • If the dark bars are going toward each other, the answer is All Real Numbers, so you would graph a darkened bar over the entire number line.
ExamplesExamples:
x - 4 ≤ 3 OR 2x > 18
3x + 1 < 4 OR 2x - 5 > 7
2x + 1 ≤ 7 OR -3x - 4 ≥ 2



G Redden

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