Presentation on theme: "Revision Linear Inequations"— Presentation transcript:
1 Revision Linear Inequations NaturalNumbersNIntegersJRationalQIrrationalIReal NumbersRComplexCAlgebraic and Graphical Solutions.By I Porter
2 IntroductionAn inequation is formed when two mathematical statements have an unequalitysign between them.Common inequality signs:> is greater than≥ is greater than or equal to< is less than≤ is less than or equal toInequations can have an infinite number of solutions.Solving inequations makes use of the following axioms of inequality for realnumbers a, b and c.If a > b , then1. a + c > b + c5. ac < bc if c < 02. a - c > b - cif c < 03. ac > bc if c > 0if c > 0Similar axioms also apply for a < b.
3 Solving Inequalities Inequations may be simplified by: 1. adding the same number to both sides.i.e. 10 > 3, then > 3 + 22. subtracting the same number to both sides.i.e. 10 > 3, then > 3 - 23. multiplying both sides by the same positive number.i.e. 10 > 3, then 10 x 2 > 3 x 2i.e. 10 > 3, then4. 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 CasesThe 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 -2i.e. 10 > 3, then <2. dividing both sides by the same negative number.
4 Graphical Solutions Algebra Graphical Number Line Solution x > 4 x 1617181920x-12-11-10-9-8-7-6-5-4-345632x < -5x ≥ -10x ≤ 18Note: ‘>’ 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.
5 ExamplesInequations 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 < 23Add 5 to both sides.b) 4(2 - x) ≥ 3x + 14Expand Brackets.4x < 28Divide both sides by 4.8 - 4x ≥ 3x + 14Subtract 3x from both sides.x < 7Open circle, arrow left.8 - 7x ≥ 14Subtract 8 from both sides.- 7x ≥ 6Divide both sides by -7.876xReverse inequality sign.Closed circle, arrow left.xAlways move ALGEBRA to the LEFT SIDE
6 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 CircleClosed CircleAdd 3 to both sides.Divide both sides by -2.Reverse inequality sign.Closed circle, arrow right.-2-11234567-3-4-5xAlways move ALGEBRA to the LEFT SIDE
7 Exercise: Solve and graph a solution for each the following: -4-5-6x432x-2-3-4x-5-6-7x67854x5-4xAlways move ALGEBRA to the LEFT SIDE