Solve Linear Inequalities - Tutorial

Solve linear inequalities: A tutorial with examples and detailed solutions. Double inequalities and inequalities with fractional expressions are also included.

We, first, review some of the properties of the inequalities.

Properties of inequalities

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

6x - 6 > 2x + 2

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

10x - 8 > 4x + 10


Example 2: Solve the inequality

2(3x + 2) - 20 > 8(x - 3)

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

-3(4x + 1) + 10 > -4(x - 3)


Example 3: Solve the double inequality

-3 < 4(x + 2) - 3 < 9

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

-1 < -2(x - 3) - 3 < 7


Example 4: Solve the inequality

(x + 2) / 3 - 2 / 5 < (-x - 1) / 3 - 1 / 6

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

(x - 2) / 4 - 2 / 7 < (-x + 3) / 7 - 1 / 2



More references and links to inequalities

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



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