**Solving Equations and Inequalities by Graphing :**

In this section, we will learn, how to solve equations and inequalities 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.

So, -3x + 20 = 5 when x = 5.

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

Hence, 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

Hence, the solutions to tBy substituting this value into the original equation, we can verify the result.

So, -3x + 20 = 5 when x = 5.

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

Hence, the solutions to the equations y = |x - 4| and aaaaaa y = 0.5x + 1 is

x = 2 and x = 10

**Example 1 : **

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

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

x < 5 or x ∈ (-∞, 5)

**Example 2 : **

Solve the following inequality by graphing :

x^{2} - 4 > 0

**Solution : **

To solve the inequality, we have to identify the values of x that make the value of the expression x^{2} - 4 greater than 0.

Graph the equation y = x^{2} - 4 by translating the parent function y = x^{2} down 4 units.

In the above graph, y > 0 when

x < -2 and x > 2

That is, x^{2} - 4 > 0 when

x < -2 and x > 2

Hence, the solution to the inequality x^{2} - 4 > 0 is

x < -2 or x > 2

or

x ∈ (-∞, 2)U(2, ∞)

After having gone through the stuff given above, we hope that the students would have understood, how to solve equations and inequalities by graphing.

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