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**Revision Linear Inequations**

Natural Numbers N Integers J Rational Q Irrational I Real Numbers R Complex C Algebraic and Graphical Solutions. By I Porter

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Introduction An inequation is formed when two mathematical statements have an unequality sign between them.Common inequality signs: > is greater than ≥ is greater than or equal to < is less than ≤ is less than or equal to Inequations can have an infinite number of solutions. Solving inequations makes use of the following axioms of inequality for real numbers a, b and c. If a > b , then 1. a + c > b + c 5. ac < bc if c < 0 2. a - c > b - c if c < 0 3. ac > bc if c > 0 if c > 0 Similar axioms also apply for a < b.

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**Solving Inequalities Inequations may be simplified by:**

1. adding the same number to both sides. i.e. 10 > 3, then > 3 + 2 2. subtracting the same number to both sides. i.e. 10 > 3, then > 3 - 2 3. multiplying both sides by the same positive number. i.e. 10 > 3, then 10 x 2 > 3 x 2 i.e. 10 > 3, then 4. dividing both sides by the same positive number. In all cases above, the direction of the inequality remains the same. Also, the above statements apply when ‘>’ is replaced with ‘<‘. Special Cases The inequality sign must be reversed when: 1. multiplying both sides by the same negative number. i.e. 10 > 3, but 10 x -2 < 3 x -2 i.e. 10 > 3, then < 2. dividing both sides by the same negative number.

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**Graphical Solutions Algebra Graphical Number Line Solution x > 4 x**

16 17 18 19 20 x -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 4 5 6 3 2 x < -5 x ≥ -10 x ≤ 18 Note: ‘>’ and ‘< ‘ use and open circle. Note: ‘≥’ and ‘≤‘ use and closed (dot) circle. You also only need to write in three (3) numbers to indicate location and order.

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Examples Inequations are solve exactly the same way as equations, with two exception as stated by the axioms (5) and (6). [ reverse the inequality when x or by a negative number ] Solve and graph a solution for the following: a) 4x - 5 < 23 Add 5 to both sides. b) 4(2 - x) ≥ 3x + 14 Expand Brackets. 4x < 28 Divide both sides by 4. 8 - 4x ≥ 3x + 14 Subtract 3x from both sides. x < 7 Open circle, arrow left. 8 - 7x ≥ 14 Subtract 8 from both sides. - 7x ≥ 6 Divide both sides by -7. 8 7 6 x Reverse inequality sign. Closed circle, arrow left. x Always move ALGEBRA to the LEFT SIDE

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**Examples: Solve and graph a solution for the following:**

b) Add 3 to both sides. Cross multiply denominators. Divide both sides by 2. Expand brackets. Subtract 5x from both sides. Open Circle Closed Circle Add 3 to both sides. Divide both sides by -2. Reverse inequality sign. Closed circle, arrow right. -2 -1 1 2 3 4 5 6 7 -3 -4 -5 x Always move ALGEBRA to the LEFT SIDE

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**Exercise: Solve and graph a solution for each the following:**

-4 -5 -6 x 4 3 2 x -2 -3 -4 x -5 -6 -7 x 6 7 8 5 4 x 5 -4 x Always move ALGEBRA to the LEFT SIDE

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Solving Inequalities We can solve inequalities just like equations, with the following exception: Multiplication or division of an inequality by a negative.

Solving Inequalities We can solve inequalities just like equations, with the following exception: Multiplication or division of an inequality by a negative.

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