The Intersection of Graphs Method can be used to find solutions for an inequality. To solve the inequality using this method, follow the steps below.

 

Intersection of Graphs Method of Solving an Inequality

This method uses graphing of functions to solve an inequality. 

STEP 1: Set Y sub 1 equal to the left side of the inequality and Y sub 2 equal to the right side of the inequality.

STEP 2: Graph Y sub 1 and Y sub 2 in the same viewing rectangle. 

STEP 3: Locate any points of intersection.  The x-values of these points correspond to points that are the boundaries where Y sub 1 > Y sub 2 or Y sub 1 < Y sub 2

 

STEP 1: Press the button, then enter the left side of the inequality forY sub 1 and the right side of the inequality forY sub 2.

STEP 2: Next press the ZOOM, then 6. The graphs of the functions in a standard window are shown below.

STEP 3: Note that there are 2 intersection points.  Now press the 2nd key, then TRACE [CALC], then select 5:intersection.


We see above that there are 2 intersections points. We must find the x-coordinates of both intersection points. If the cursor is not near the leftmost intersection point, move it close to the point of intersection using the ARROW keys and then press ENTER.


 When Second curve? appears press ENTER again. When Guess? appears press ENTER again.


The last screen above shows that one of the critical x values is x = -0.317. Now we must find the other intersection point.

Repeating STEP 2: Next press the GRAPH. The graphs of the functions in a standard window are shown below.

Repeating STEP 3: Now press the 2nd key, then TRACE [CALC], then select 5:intersection.


We see above that there are 2 intersections points. We must find the x-coordinates of both intersection points. If the cursor is not near the rightmost intersection point, move it close to the point of intersection using the ARROW keys and then press ENTER.


 When Second curve? appears press ENTER again. When Guess? appears press ENTER again.

The second critical x value is x = 6.317. Both critical values, x = -0.317 and x = 6.317, are shown on the number line below.

Now we must decide which one(s) of the three resulting intervals are part of the solution set. To determine that, use the table function of the calculator. First press 2nd WINDOW[TBLSET] to get the TABLE SETUP window. Enter -5 for TblStart and 1 for . Then press 2nd GRAPH[TABLE] to view the table of x and y values. Scroll down to see more values.

   

The inequality that we are solving is of the form, Y sub 1 > Y sub 2. For the x values less than the smaller critical value, x = -0.317, we see that Y sub 1 < Y sub 2. These x's are not in the solution set. However, for x values greater than -0.317, Y sub 1 > Y sub 2 until the x values become greater than the second critical value, 6.317. For x's greater than 6.317, we find that once again Y sub 1 < Y sub 2. Thus, the solution to the inequality, , is -0.317 < x < 6.317, or in interval notation, (-0.317, 6.317). See the graph of the solution set below.

Note that the solution to the inequality, , would be obtained through the same steps up to the selection of the solution set. Since Y sub 1 < Y sub 2 when
x < -0.317 or x > 6.317, the solution set would be graphed as seen below.

 



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