Presentation on theme: "Revision Linear Inequations"— Presentation transcript:
1Revision Linear Inequations NaturalNumbersNIntegersJRationalQIrrationalIReal NumbersRComplexCAlgebraic and Graphical Solutions.By I Porter
2IntroductionAn 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.
3Solving 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.
4Graphical 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.
5ExamplesInequations 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
6Examples: 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
7Exercise: 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