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### High School Mathematics10.7 Solving Inequations

Properties of Inequalities: less than or equal to £ greater than or equal to ³
 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 ³ b, then a + c ³ b + c Subtraction Property of Inequality Subtracting 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 ³ b, then a - c ³ 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 ³ b AND c is positive, then a/c ³ 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 ³ 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 ³ 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 Öa > Öb Reciprocal On both sides of the inequality changes the inequality sign 1/a > 1/b a < b
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 ³ b, then a + c ³ b + c
If a £ b, then a - c £ b - c
If a ³ b, then a - c ³ 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 ³ b AND c is positive, then a/c ³ 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 ³ bc
If a £ b AND c is negative, then a/c ³ 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: Ö9 > Ö6
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 4x - 7 < 21.x < 12/7x < 12x < 7x < 7/12 Q 2: Solve 3x + 2 < 15.x < 3x < 3/13x < 13/3x < 13 Q 3: Solve -7b + 19 < -16b > 5b < -5b > -5b < 5 Q 4: Solve (3x - 7)/4 > (4x - 8)/3.x > 11x > 7/11x < 11/7x > 7 Q 5: Solve 7x - 8 < 15.x < 7/23x < 23/7x < 23x < 7 Q 6: Solve 4(x + 3) < 3(x - 3).x < -15x < 21x < 15x < -21 Q 7: Solve 5(2x + 3) < 3(2x - 5).x < -15/2x < 2/15x < -2/15x < 15/2 Q 8: Solve 8(t + 2) - 3(t - 4) < 5(t - 7) + 8 solution is empty setsolution set is {t|t > 27}solution set is {t|t < 28}solution set is {t|t > 28} Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!