SOLVING EQUATIONS AND INEQUALITIES BY GRAPHING

Solving Equations by Graphing

Example 1 :

Solve the following equation by graphing : 

-3x + 20  =  5 

Solution :

An equation is a statement that two expressions are equal. The values of x that make the equation true are the solutions. 

To have an equation by graphing, write two new equations by setting y equal to each in the original equation. 

-3x + 20  =  5

y  =  -3x + 20     y  =  5   

Graph the two equations and identify the points of intersection. These points will have x-values that produce the same y-values for both expressions. 

Each of the x-values is a solution to the original equation. 

Graph y  =  -3x + 20 and y  =  5. 

It appears that y = -3x + 20 and y = 5 intersect at

x  =  5

By substituting this value into the original equation, we can verify the result. 

That is,

-3(5) + 20  =  5

-15 + 20  =  5

5  =  5

This is the only point on the graph where the value of the functions y = -3x + 20 and y = 5 are equal. 

So, the solution to the equation -3x + 20  =  5 is

x  =  5

Example 2 : 

Solve the following equation by graphing : 

|x - 4|  =  0.5x + 1 

Solution :

As we have done in example 1 above, write two new equations by setting y equal to each in the original equation. 

|x - 4|  =  0.5x + 1

y  =  |x - 4|     y  =  0.5x + 1   

Graph the two equations and identify the points of intersection. These points will have x-values that produce the same y-values for both expressions. 

It appears that y = |x - 4| and y = 0.5x + 1 intersect at

x  =  2 and x  =  10

By substituting these values into the original equation, we can verify the result. 

That is, 

So, the solutions to the equation |x - 4|  =  0.5x + 1 are

x  =  2 and x  =  10

Solving Inequalities by Graphing

Example 3 : 

Solve the following inequality by graphing : 

 -3x + 20  >  5

Solution :

To solve the inequality, we have to identify the values of x that make the value of the expression -3x + 20 greater than 5. 

To have an inequality by graphing, write two new equations by setting y equal to each in the original inequality. 

-3x + 20  >  5

y  =  -3x + 20     y  =  5   

Graph the two equations and identify the points of intersection. These points will have x-values that produce the same y-values for both expressions. 

Graph y  =  -3x + 20 and y  =  5. 

From the point of intersection, it is clear that y  =  5 when x  =  5.

In the above graph, y > 5 when x < 5. 

That is, 

-3x + 20 > 5 when x < 5

So, the solution to the inequality -3x + 20 > 5 is 

x < 5 or x ∈ (-∞, 5)

Example 4 : 

Solve the following inequality by graphing : 

 x² - 4 > 0

Solution : 

To solve the inequality, we have to identify the values of x that make the value of the expression x² - 4 greater than 0. 

Graph the equation y  =  x² - 4 by translating the parent function y  =  x² down 4 units. 

In the above graph, y > 0 when

x < -2 and x > 2

That is, x² - 4 > 0 when

x < -2 and x > 2

So, the solution to the inequality x² - 4 > 0 is 

x < -2 or x > 2

or

x ∈ (-∞, 2)U(2, ∞)

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