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Solving Compound Inequalities Equations

Key Concepts

  • Define a compound inequality
  • Solve a compound inequality involving “or”
  • Solve a compound inequality involving “and”
  • Use compound inequality to solve problems

Compound Inequality 

  • A compound inequality is made of two or more inequalities. 

Step 1: Write an inequality to represent the solutions shown in each part of the graph. 

Step 2: Draw the compound inequality.  

The compound inequality describes the graph of x ≤ -3 or x > 2. 

The compound inequality describes the graph of x ≤ -3 or x > 2.

Solve compound inequality including “or” 

  • When a compound inequality with an “or” is given, the graph shows all points that appear in either of the solutions above. 

Example: Solve 2x-5 > 3 or -4x+7 < -25 

Step 1: Solve and graph 2x-5 > 3 

              2x-5+5 > 3+5 

              2x > 8 

2x/2 > 8/2

               x > 4 

Step 2: Solve and graph 4x-7 < 25 

              4x+7–7 < 25+7 

               4x < 32 

4x/4 < 32/4

                x < 8 

Solve compound inequality including "or" 

The solutions can be 5, 6, 7. 

Solving compound inequality including “and” 

  • When a compound inequality with an “and” is given, the final graph shows all points that appear in both solutions above. 

Example: Solve -12 ≤ 7x+9 < 16 

Step 1: Solve and graph -12 ≤ 7x+9  

              -12 – 9 ≤ 7x + 9 – 9 

              -21 ≤ 7x 

−217 ≤ 7x/7

             -3 ≤ x 

Step 2: Solve and graph 7x+9 < 16  

              7x+9-9 < 16-9 

               7x < 7 

7x/7 < 7/7

                x < 1 

Solving compound inequality including "and" 

The solution is x-3 and  x<1, or -3≤x<1 

Exercise

  • Solve 12<2x<28
  • The compound function that represents the graph is _________.
exercise 2
  • Write the compound inequality that represents the area A of the rectangle if 35≥A25
exercise 4
  • Find the area of the right-angled triangle if the height is 5 units and the base is x units, given that the area of the triangle lies between 10 and 35 sq. units.
  • Write an inequality that represents the quantity that is greater than 18 but less than or equal to 27

Concept Map

Concept Map

What have we learned

  • A compound equality is made up of two or more inequalities.

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