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Fundamentals
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2. 1.11 Solving Equations and Inequalities Graphically
Copyright © Cengage Learning. All rights reserved.

3. Objectives Solving Equations Graphically
Solving Inequalities Graphically

4. Solving Equations and Inequalities Graphically
Sometimes an equation or inequality may be difficult or impossible to solve algebraically. In this case we use the graphical method.
In this method we view x as a variable and sketch an appropriate graph. We can then obtain an approximate solution from the graph.

5. Solving Equations Graphically

6. Solving Equations Graphically
To solve a one-variable equation such as 3x – 5 = 0 graphically, we first draw a graph of the two-variable equation y = 3x – 5 obtained by setting the nonzero side of the equation equal to a variable y.
The solutions of the given equation are the values of x for which y is equal to zero. That is, the solutions are the x-intercepts of the graph.

7. Solving Equations Graphically
The following describes the method.

8. Solving Equations Graphically
The advantage of the algebraic method is that it gives exact answers. Also, the process of unraveling the equation to arrive at the answer helps us to understand the algebraic structure of the equation.
On the other hand, for many equations it is difficult or impossible to isolate x.
The graphical method gives a numerical approximation to the answer.
This is an advantage when a numerical answer is desired.

9. Example 1 – Solving a Quadratic Equation Algebraically and Graphically
Find all real solutions of the quadratic equation. Use the algebraic method and the graphical method.
(a) x2 – 4x + 2 = (b) x2 – 4x + 4 = 0 (c) x2 – 4x + 6 = 0
Solution 1: Algebraic You can check that the Quadratic Formula gives the following solutions.
(a) There are two real solutions, x = and x = 2 – .
(b) There is one real solution, x = 2.

10. Example 1 – Solution
cont’d
(c) There is no real solution. (The two complex solutions are x = and x = 2 – )

11. Example 1 – Solution
cont’d
Solution 2: Graphical We use a graphing calculator to graph the equations
y = x2 – 4x + 2,
y = x2 – 4x + 4, and
y = x2 – 4x + 6 in Figure 1.
(a) y = x2 – 4x + 2
(b) y = x2 – 4x + 4
(c) y = x2 – 4x + 6
Figure 1

12. Example 1 – Solution
cont’d
By determining the x-intercepts of the graphs, we find the following solutions.
(a) The two x-intercepts give the two solutions x  0.6 and x  3.4.
(b) The one x-intercept gives the one solution x = 2.
(c) There is no x-intercept, so the equation has no real solutions.

13. Solving Inequalities Graphically

14. Solving Inequalities Graphically
To solve a one-variable inequality such as 3x – 5  0 graphically, we first draw a graph of the two-variable equation y = 3x – 5 obtained by setting the nonzero side of the inequality equal to a variable y.
The solutions of the given inequality are the values of x for which y is greater than or equal to 0. That is, the solutions are the values of x for which the graph is above the x-axis.

15. Solving Inequalities Graphically

16. Example 5 – Solving an Inequality Graphically
Solve the inequality 3.7x x – 1.9  2.0 – 1.4x.
Solution: We use a graphing calculator to graph the equations
y1 = 3.7x x – and y2 = 2.0 – 1.4x
The graphs are shown in Figure 5.
y1 = 3.7x x – 1.9
y2 = 2.0 – 1.4x
Figure 5

17. Example 5 – Solution
cont’d
We are interested in those values of x for which y1  y2; these are points for which the graph of y2 lies on or above the graph of y1.
To determine the appropriate interval, we look for the x-coordinates of points where the graphs intersect.
We conclude that the solution is (approximately) the interval [–1.45, 0.72].