Solve Linear Inequalities - Tutorial

Let a, b and c be real numbers.

1. Transitive Property
   If a < b and b < c then a < c
   
   
2. Addition Property
   If a < b then a + c < b + c
   
   
3. Subtraction Property
   If a < b then a - c < b - c
   
   
4. Multiplication Property
   1. If a < b and c is positive then c*a < c*b
      
   2. If a < b and c is negative c*a > c*b
   Note:
   1. If each inequality sign is reversed in the above properties, we obtain similar properties.
   2. If the inequality sign < is replaced by <= ( less than or equal) or the sign > is replaced by >= ( greater than or equal ), we also obtain similar properties.

Example 1: Solve the inequality

Solution to Example 1:

• Given
  6x - 6 > 2x + 2
  
  
• Add 6 to both sides and simplify (Property 2 above)
  6x > 2x + 8
  
  
• Subtract 2x to both sides and simplify ( Property 3 above)
  4x > 8
  
  
• Multiply both sides by 1/4; and simplify ( Property 4-i above)
  x > 2
  
  
• Conclusion
  The solution set consists of all real numbers in the interval (2 , + infinity).
  

Matched Exercise: Solve the inequality

Example 2: Solve the inequality

Solution to Example 2:

• Given
  2(3x + 2) -20 > 8(x - 3)
  
  
• Multiply factors and group like terms
  6x + 4 -20 > 8x - 24
  6x - 16 > 8x - 24
  
  
• Add 16 to both sides and simplify ( Property 2 above)
  6x > 8x - 8
  
  
• Subtract 8x to both sides and simplify ( Property 3 above)
  -2x > -8
  
  
• Multiply both sides by -1/2 and REVERSE (-1/2 is negative) the inequality sign and simplify ( Property 4-ii above)
  x < 4
  
  
• Conclusion The solution set consists of all real numbers in the interval (- infinity , 4)

Matched Exercise: Solve the inequality

Example 3: Solve the double inequality

Solution to Example 3:

• Given
  -3 < 4(x + 2) - 3 < 9
  
  
• Multiply factors and group like terms
  -3 < 4x + 8 - 3 < 9
  -3 < 4x + 5 < 9
  
  
• Subtract 5 to all three terms and simplify
  -3 - 5 < 4x + 5 - 5 < 9 - 5
  -8 < 4x < 4
  
  
• Divide all three terms by 4
  -2 < x < 1
  
  
• Conclusion
  The solution set consists of all real numbers in the interval (- 2 , 1)
  

Matched Exercise: Solve the double inequality

Example 4: Solve the inequality

Solution to Example 4:

• Given
  (x + 2) / 3 - 2 / 5 < (-x - 1) / 3 - 1 / 6
  
  
• Multiply all terms by 30, the LCD
  30*(x + 2) /3 - 30*2 / 5 < 30(-x - 1) / 3 - 30*1 / 6
  
  
• simplify
  10(x + 2) - 6*2 < 10(-x - 1) - 5
  
  
• Multiply factors and group like terms
  10x + 20 - 12 < -10x - 10 - 5
  10x + 8 < -10x -15
  
  
• Subtract 8 to both sides and simplify
  10x < -10x - 23
  
  
• Add 10x to both sides and simplify
  20x < -23
  
  
• Divide both sides by 20
  x < -23 / 20
  
  
• Conclusion
  The solution set consists of all real numbers in the interval
  (- infinity , -23/20).
  

Matched Exercise: Solve the inequality

More references and links on how to Solve Equations, Systems of Equations and Inequalities.