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2.18 Graphical solution of inequalities Introduction Graphs can be used to solve inequalities. This leaflet illustrates how. 1. Solving inequalities We start with a very simple example which could be solved very easily using an algebraic method. Example Solve the inequality x + 3 > 0. Solution We seek values of x which make x + 3 positive. There are many such values, e.g. try x = 7 or x = −2. To find all values first let y = x + 3. Then the graph of y = x + 3 is sketched as shown below. From the graph we see that the y coordinate of any point on the line is positive whenever x has a value greater than −3. That is, y > 0 when x > −3. But y = x + 3, so we can conclude that x + 3 will be positive when x > −3. We have used the graph to solve the inequality. y y is positive when x is greater than -3 y y =x+3 y is positive when x is less than -1 and when x is greater than 3 3 -3 x -1 3 x Example Solve the inequality x2 − 2x − 3 > 0. Solution We seek values of x which make x2 − 2x − 3 positive. We can find these by sketching a graph of y = x2 −2x−3. To help with the sketch, note that by factorising we can write y as (x+1)(x−3). The graph will cross the horizontal axis when x = −1 and when x = 3. The graph is shown www.mathcentre.ac.uk 2.18.1 c Pearson Education Ltd 2000 above on the right. From the graph note that the y coordinate of a point on the graph is positive when either x is greater than 3 or when x is less than −1. That is, y > 0 when x > 3 or x < −1 and so: x2 − 2x − 3 > 0 when x > 3 or x < −1 Example Solve the inequality (x − 1)(x − 2)(x − 3) > 0. Solution We consider the graph of y = (x − 1)(x − 2)(x − 3) which is shown below. It is evident from the graph that y is positive when x lies between 1 and 2 and also when x is greater than 3. The solution of the inequality is therefore 1 < x < 2 and x > 3. y y is positive when x lies between 1 and 2 and when x is greater than 3. 2 1 Example For what values of x is x+3 x−7 x 3 positive ? Solution x+3 The graph of y = x−7 is shown below. We can see that the y coordinate of a point on the graph is positive when x < −3 or when x > 7. x+3 >0 x−7 when x < −3 or when x>7 y y is positive when x is less than -3 and when x is greater than 7. 1 -5 5 10 x x = -3 x=7 For drawing graphs like this one a graphical calculator is useful. www.mathcentre.ac.uk 2.18.2 c Pearson Education Ltd 2000