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(a) How to solve the quadratic inequalities in one unknown by using graphical method? 12. Inequalities and Linear Programming For y > 0, graph y = ax 2.

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Presentation on theme: "(a) How to solve the quadratic inequalities in one unknown by using graphical method? 12. Inequalities and Linear Programming For y > 0, graph y = ax 2."— Presentation transcript:

1 (a) How to solve the quadratic inequalities in one unknown by using graphical method? 12. Inequalities and Linear Programming For y > 0, graph y = ax 2 +bx+c are and β. the solution of the inequality is The x-intercepts of the quadratic above the x-axis, y x y = ax 2 +bx+c β O or x < x >β Using the graph of y = ax 2 +bx+c, find the range of values of x by reading the points lying above the x-axis. (i) a > 0, solve ax 2 +bx+c > 0. i.e. the blue curve.

2 (i) a > 0, For y < 0, the solution of the inequality is y x y = ax 2 +bx+c β O < x <β Using the graph of y = ax 2 +bx+c, find the range of values of x by reading the points lying below the x-axis. solve ax 2 +bx+c < Inequalities and Linear Programming graph y = ax 2 +bx+c are and β. The x-intercepts of the quadratic below the x-axis, (a) How to solve the quadratic inequalities in one unknown by using graphical method? i.e. the orange curve.

3 (ii) a 0. The range of values of x is the solution of the inequality is The quadratic graph is vertically above the x-axis, For y > 0, O β y x y = ax 2 +bx+c < x <β inverted compared with (i), opposite to the case a > 0. Using the graph of y = ax 2 +bx+c, find the range of values of x by reading the points lying above the x-axis. 12. Inequalities and Linear Programming (a) How to solve the quadratic inequalities in one unknown by using graphical method? i.e. the blue curve.

4 (ii) a < 0, the solution of the inequality is below the x-axis, For y < 0, O β y x y = ax 2 +bx+c x < orx >β Using the graph of y = ax 2 +bx+c, find the range of values of x by reading the points lying below the x-axis. solve ax 2 +bx+c < Inequalities and Linear Programming The quadratic graph is vertically inverted compared with (i), The range of values of x is opposite to the case a > 0. (a) How to solve the quadratic inequalities in one unknown by using graphical method? i.e. the orange curve.

5 (i) Solve the inequality x 2 - 3x - 10 > 0. (b) For algebraic method, how to use the graphs or tables to find the solutions of the quadratic inequalities in one unknown?E.g. y x y = x 2 - 3x+10 O The graph above the x-axis stand for y > 0.Graphical Consider x 2 - 3x - 10 = 0, (x+2)(x - 5) = 0 x = - 2 or x = 5 a = 1, i.e. a > 0 According to the above result, x > 5 x < - 2 or we can sketch the graph. 12. Inequalities and Linear Programming

6 (i) Solve the inequality x 2 - 3x - 10 > 0.E.g. Tabular x 5 x < < x < 5-2 < x < 5 x > 5 x+2 x-5x-5 (x+2)(x - 5) x< - 2 x+2< x+2<0 x< - 2 x - 5< x - 5<0 x+2<0 x - 5<0 (x+2)(x - 5)>0 - 20 - 20 x - 5<0 (x+2)(x - 5)<0 x>5 x+2>5+2 x+2>7 x+2>0 x>5 x - 5>5 - 5 x - 5>0 x+2>0 x - 5>0 (x+2)(x - 5)>0 12. Inequalities and Linear Programming (b) For algebraic method, how to use the graphs or tables to find the solutions of the quadratic inequalities in one unknown? Consider x 2 - 3x - 10 = 0, (x+2)(x - 5) = 0 x = - 2 or x = 5

7 (ii) Solve the inequality x 2 +5x - 6 < 0. Consider x 2 +5x - 6 = 0, (x+6)(x - 1) = 0 x = - 6 or x = 1 a = 1, i.e. a > 0 According to the above result, we can sketch the graph. y x y = x 2 +5x O - 6 < x < Inequalities and Linear Programming Graphical E.g. (b) For algebraic method, how to use the graphs or tables to find the solutions of the quadratic inequalities in one unknown? The graph below the x-axis stand for y < 0.

8 (ii) Solve the inequality x 2 +5x - 6 < 0.E.g. - 6 < x < 1 x < < x < 1-6 < x < 1 x > 1 x+6 x-1x-1 (x+6)(x - 1) x< - 6 x+6< x+6<0 x< - 1 x - 1< x - 1< - 7 x - 1<0 x+6<0 x - 1<0 (x+6)(x - 1)>0 - 60 - 60 x - 1<0 (x+6)(x - 1)<0 x>1 x+6>1+6 x+6>7 x+6>0 x>1 x - 1>1 - 1 x - 1>0 x+6>0 x - 1>0 (x+6)(x - 1)>0 12. Inequalities and Linear Programming Tabular (b) For algebraic method, how to use the graphs or tables to find the solutions of the quadratic inequalities in one unknown? Consider x 2 +5x - 6 = 0, (x+6)(x - 1) = 0 x = - 6 or x = 1

9 the solution of the inequality is < x <β. the solution of the inequality is x β. Solving Quadratic Inequalities Easy Memory Tips Otherwise, When the quadratic function and a (the coefficient of x 2 ) are both larger than zeroor smaller than zero, 12. Inequalities and Linear Programming ax 2 +bx+c > 0 a < 0 ax 2 +bx+c > 0 a > 0 ax 2 +bx+c < 0 a < 0 ax 2 +bx+c < 0 a > 0


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