Chapter 3.3

Graphical Solutions of Equations

In Sections 2.1 and 2.2 we learned how to solve equations algebraically.  For example:

Example 1.  Solve  for x.

So we obtained

However, in real world examples we are usually not presented with numbers such as this.  How can we use the technology (i.e. the graphing calculator) to help solve, not only problems of this type, but ones that are a bit more challenging?

I will try to walk you through step-by-step.  The book does a really good job on pages 156-161 if you would rather read it.

Let’s use the TI-83 to help us solve example 1:

Example 2. Solve for x: 

Step 1.  Enter  in for [Y1] and  in for [Y2].  Note:  You must use the  key to enter the x variable.

Step 2.  It is now time to graph the equation.  There are several ways to do this.  Since we have relatively small numbers, let’s use the [zoom]  and then go to [ZStandard].  See the screen of the left.  Just hit the [ENTER] key.

Step 4.  You should get the graph on the left.

Step 5.  We are going top find the x-value  where the two lines intersect.  To do so hit the [2nd] [calc].  You should get the screen on the left.  Go down to number 5: intersection and hit the [ENTER] key.  (Note:  you may also just hit the 5)

Step 6.  It is going to ask for you to identify the first curve.  Because we only have two curves, just hit the [ENTER] key three times.  In later sections we will explain this further.

Step 7.  After hitting the [ENTER] key three times, you should get the graph on the left.  Note:  You have not found the intersection until you see the word “Intersection”.  In our problem the answer is x=-2.  This confirms our answer we obtained in example 1.

Quick check:

Solve using the TI-83.  Check your answer by solving algebraically.

Answer:

1) 

So

2) 

So

Inequalities

Let’s revisit inequalities.  In this section we will not solve them algebraically—we will just inspect the graph to determine the solution interval.

Example 3.  Determine the solution set of the inequality given the graph below.

As in the first example put the left side in for Y1 and the right side in for Y2.  I am using color just to make it easier to follow.  Once you have it graphed, determine x-value of the intersection like you did in example 1. 

In this problem the x-value of the intersection is .  But we are asked to find more than just the intersection.   We must find “Where the red line is below the blue line?”.  We note that for values to the left of 3 the red line is below the blue line.  So our answer is, in interval notation, .

Example 4.  Solve the inequality by using the TI-83.

Again, I will use the color graphs to illustrate.

Here we are trying to find where the red line is above the blue line.  Finding the x-coordinate of the intersection we obtain .

From inspection, we see that the red line is above the blue line to the left of 1.  Therefore, we may write the answer in interval notation as .

Here is another way to do it using the TI-83. This way involves many keystrokes and you must have the equations given to you. Sometimes we are asked to look at only the graphs without being given the equations.

Quick check:

Determine the solution set for the inequality below.  You will need to use the TI-83 to graph.

2)  Given the graph below, give the interval where (for what x-values) the red line is below the blue line. (Note that in this one you have to understand it by looking at the graph.)

Answer:

1)

Note: You may have to use the trace button to determine which line is which equation.

2)