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High School Mathematics
10.7 Solving Inequations

Properties of Inequalities: less than or equal to greater than or equal to
Addition Property of Inequality
Adding both sides of an inequality with a
positive number does not change the inequality sign If a < b, then a + c < b + c
If a b, then a + c b + c
If a b, then a + c b + c
Subtraction Property of Inequality
Subtracting both sides of an inequality with a
positive number does not change the inequality signIf a < b, then a - c < b - c
If a b, then a - c b - c
If a b, then a - c b - c
Multiplication Property of Inequality
Mulitplying both sides of the inequality with a
positive number does not change the inequality sign If a < b AND c is positive, then ac < bc
If a b AND c is positive, then ac bc
Division Property of Inequality positive number does not change the inequality sign If a < b AND c is positive, then a/c < b/c
If a b AND c is positive, then a/c b/c
Multiplication Property of Inequality negative number changes the inequality sign If a < b AND c is negative, then ac > bc
If a b AND c is negative, then ac bc
Division Property of Inequality negative number changes the inequality sign If a < b AND c is negative, then a/c > b/c
If a b AND c is negative, then a/c b/c
If both sides of an inequality are positive and n is a positive integer n-th power or n-th root does not change the inequality a > b
a2 > b2
a > b
Reciprocal On both sides of the inequality changes the inequality sign 1/a > 1/b
a < b
  • Addition/Subtraction Property of Inequality:
    Adding or subtracting a positive number from both sides of an inequality does not change the inequality sign.
    If a < b, then a + c < b + c
    If a < b, then a - c < b - c
    If a b, then a + c b + c
    If a b, then a + c b + c
    If a b, then a - c b - c
    If a b, then a - c b - c
  • Multiplication/Division Property of Inequality:
    Multiplying and dividing both sides of an inequality by a positive number does not change the inequality sign. This is not true for a negative number b.
    If a < b AND c is positive, then ac < bc
    If a < b AND c is positive, then a/c < b/c
    If a b AND c is positive, then ac bc
    If a b AND c is positive, then a/c b/c
  • Multiplying or dividing both sides of an inequality by a negative number changes the inequality sign.
    If a < b AND c is negative, then ac > bc
    If a < b AND c is negative, then a/c > b/c
    If a b AND c is negative, then ac bc
    If a b AND c is negative, then a/c b/c
  • If both sides of an inequality are positive and n is a positive integer, then the inequality formed by the n-th power or n-th root of both sides does not change the inequality.
    Example: 9 > 6
    92 > 62
    81 > 36 That is still true.
    Example: 9 > 6
    3 > 2.45
  • Taking the reciprocal on both sides of the inequality changes the inequality sign.
    Example:
    1/2 > 1/4
    Taking reciprocal both sides changes the inequality
    2 < 4
Example:
Solve 5x + 3 < 10.
Solution:
Given that 5x + 3 < 10.
Add -3 on both sides, we get
5x + 3 - 3 < 10 - 3
5x < 7
Dividing both sides by 5, we get
5x/5 < 7/5
x < 7/5
Therefore, any number x < 7/5 is a solution.

Example:
Solve 3d - 2(8d - 9) > -2d - 4
Solution:
3d - 2(8d - 9) > -2d - 4 Original inequality
3d - 16d + 18 > -2d - 4 Distributive property
- 13d + 18 > -2d - 4 Combining like terms
- 13d + 18 + 13d > -2d - 4 + 13d Adding 13d both sides
18 > 11d - 4 Simplify
18 + 4 > 11d - 4 + 4 Adding 4 each side
22 > 11d Simplify
22/11 > 11d/11 Divide each side by 11
2 > d Simplify
That is d > 2
Therefore the solution set is {d|d < 2}


Directions: Solve the following inequations. Also write at least 10 examples of your own.
Q 1: Solve (3x - 7)/4 > (4x - 8)/3.
x < 11/7
x > 7/11
x > 7
x > 11

Q 2: Solve 3x + 2 < 15.
x < 13
x < 3/13
x < 3
x < 13/3

Q 3: Solve -7b + 19 < -16
b > 5
b > -5
b < 5
b < -5

Q 4: Solve 8(t + 2) - 3(t - 4) < 5(t - 7) + 8
solution set is {t|t < 28}
solution set is {t|t > 27}
solution set is {t|t > 28}
solution is empty set

Q 5: Solve 4x - 7 < 21.
x < 12
x < 12/7
x < 7
x < 7/12

Q 6: Solve (7x - 8)/2 < (9x - 6)/3.
x < 6
x < -4
x < 4
x < -6

Q 7: Solve 5(x + 5) < 3(x + 5).
x < 20
x < 15
x < -5
x < -15

Q 8: Solve 5(2x + 3) < 3(2x - 5).
x < 15/2
x < -2/15
x < -15/2
x < 2/15

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Question 10: This question is available to subscribers only!


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