Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
How to solve inequalities. There are a number of different types of inequalities that we can solve the following notes show the method used to solve each type of inequality problem. Type 1: “Linear inequalities” The method we use to solve “linear inequalities” is the same method that is used to solve any linear equation. We will therefore add, subtract, multiply or divide both sides until you get a single variable x on its own. The main differences in solving an inequality is that if you multiply or divide both sides by a negative number then the inequality symbol will reverse direction and the second difference is that your solution will not be a single value of x but an interval of values. For example, when you solve an equation you typically get x = 2 but when you solve an inequality the solution will be an interval of values such as x > 2 or x 2. Also notice that when we give a solution we do not just stop and just say that the solution is x 4 but we also give the solution in interval notation. You must express all solutions to inequalities in interval notation unless you are told otherwise. Example 1: 2x + 3 < 2x < x < 21 18 9 Example 2: 4x – 3 4x x 17 20 5 Example 3: 5 – 2x – 2x x 13 8 –4 Example 4: 4 – 3x – 3x 2x x 12 – 5x 8 – 5x 8 4 Example 5: 4 – 3x > – 3x > 2x > x – 12 – 5x – 16 – 5x – 16 –8 Example 6: 10 – 3(x – 6) 10 – 3x + 18 28 – 3x – 3x – 4x x > > > > < ( x–2 x–2 x – 30 – 30 7.5 Notice the reversing of the symbol into Notice the reversing of the symbol into ) Type 2: “Simple Quadratic inequalities” The method we use to solve “Simple linear inequalities” is to Foil out the expressions and then to remove the x2 terms when this is done we will be left with a straight forward linear inequality and we will solve it in the usual way. To work with Simple Quadratic inequalities we need to remember the technique called F.O.I.L First Outer Inner Last Outer First For example (2x + 11)(x – 3) = Inner 2x2 – 6x + 11x F O I – 33 L Last Example 1: Example 2: Example 3: (x +3)(x – 6) x2 – 6x + 3x – 18 x2 – 3x – 18 – 3x – 18 – 3x –x x < < < < < < > x(x – 15 ) x2 – 15x x2 – 15x x2 – 15x – 15x – 9x x (2x +1)(3x – 1) 6x2 – 2x + 3x – 1 6x2 + x – 1 x –1 x 12x x < < < < < < < (x +1)(x – 3) x2 – 3x + x – 3 x2 – 2x – 3 – 2x – 3 – 2x + 15 15 – 15 Using F.O.I.L Collect like terms Subtract x2 from both sides Add 18 to both sides Add 2x to both sides Divide both sides by – 1 (x – 3)2 (x – 3)(x – 3) x2 – 3x – 3x + 9 x2 – 6x + 9 – 6x + 9 9 –1 Use the Distributive law Using F.O.I.L Collect like terms Subtract x2 from both sides Add 6x to both sides Divide both sides by – 9 (6x +1)(x – 2) 6x2 – 12x + x – 2 6x2 – 11x – 2 – 11x – 2 – 11x – 1 –1 Using F.O.I.L Collect like terms Subtract 6x2 from both sides Add 1 to both sides Add 11x to both sides Divide both sides by 2