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

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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 less than ≥ is greater than or equal to ≤ 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 2.a - c > b - c 3.ac > bc if c > 0 4. if c > 0 5.ac < bc if c < 0 6. 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. 2. subtracting the same number to both sides. 3. multiplying both sides by the same positive number. 4. dividing both sides by the same positive number. i.e. 10 > 3, then > i.e. 10 > 3, then > i.e. 10 > 3, then 10 x 2 > 3 x 2 i.e. 10 > 3, then 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. 2. dividing both sides by the same negative number. i.e. 10 > 3, but 10 x -2 < 3 x -2 i.e. 10 > 3, then <

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Graphical Solutions AlgebraGraphical Number Line Solution x > 4 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 x x x x

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x 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 4x < 28 x < x b) 4(2 - x) ≥ 3x x ≥ 3x x ≥ x ≥ 6 Add 5 to both sides. Divide both sides by 4. Open circle, arrow left. Expand Brackets. Subtract 3x from both sides. Subtract 8 from both sides. Divide both sides by -7. Reverse inequality sign. Closed circle, arrow left. Always move ALGEBRA to the LEFT SIDE

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Examples: Solve and graph a solution for the following: a) x Cross multiply denominators. Expand brackets. Subtract 5x from both sides. Add 3 to both sides. Divide both sides by -2. Reverse inequality sign. Closed circle, arrow right. Always move ALGEBRA to the LEFT SIDE b) Add 3 to both sides. Divide both sides by Open CircleClosed Circle

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Exercise: Solve and graph a solution for each the following: x 432 x x x x 05-4 x Always move ALGEBRA to the LEFT SIDE

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