Graphical representation of inequalities

Graphical Representation of Inequalities

In this section, you will learn how to graphically represent inequalities on a number line and in the Cartesian plane. This is crucial for understanding the solutions to inequalities and their implications.

Understanding Inequalities

An inequality is a mathematical statement that compares two expressions. The most common symbols used in inequalities are:

• <: less than
• >: greater than
• <=: less than or equal to
• >=: greater than or equal to

For example, the inequality x > 3 means that x can be any number greater than 3.

Graphing Linear Inequalities

To graph a linear inequality, you first graph the related linear equation. For example, consider the inequality:

x + 2 > 4

First, solve the inequality for x:

x > 4 - 2

x > 2

Next, graph the equation x = 2 as a vertical line on the Cartesian plane. Since the inequality is strict (greater than, not greater than or equal to), use a dashed line to indicate that points on the line are not included in the solution.

Now, you need to shade the region of the graph that satisfies the inequality. Since x must be greater than 2, shade to the right of the line. The graph will look like this:

    |          Shaded Region (x > 2)    
    |          
    |          
    |          
    |          
    |          
    |          
----|----
    2          
    |          
    |          

Graphing Compound Inequalities

1 < x < 5

This means that x is greater than 1 and less than 5. To graph this, you will graph both inequalities separately:

1. Graph the line x = 1 using a solid line (since it is less than or equal to).
2. Graph the line x = 5 using a solid line (since it is greater than or equal to).

After graphing, you will shade the region between the two lines:

    |          Shaded Region (1 < x < 5)    
    |          
    |          
    |          
    |          
    |          
----|----
    1          5
    |          

Graphing Quadratic Inequalities

Quadratic inequalities are inequalities that involve a quadratic expression. For example:

x^2 - 4 < 0

First, solve the related equation:

x^2 - 4 = 0

This factors to (x - 2)(x + 2) = 0, giving the solutions x = 2 and x = -2. These points divide the number line into intervals. Next, test each interval to see where the inequality holds:

• Test x = 0 (between -2 and 2): 0^2 - 4 = -4 < 0 (true)
• Test x = -3 (to the left of -2): (-3)^2 - 4 = 5 > 0 (false)
• Test x = 3 (to the right of 2): 3^2 - 4 = 5 > 0 (false)

From this, we see that the solution to the inequality is the interval (-2, 2). Graph the parabola y = x^2 - 4, using a dashed line since the inequality is strict:

    |          Shaded Region (x^2 - 4 < 0)    
    |          
    |          
    |          
    |          
    |          
----|----
    -2          2
    |          

Graphing Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value symbols. For example:

|x - 3| > 2

This means that the distance between x and 3 is greater than 2. To solve this, rewrite it as two separate inequalities:

1. x - 3 > 2
2. x - 3 < -2

Solving these gives:

1. x > 5
2. x < 1

Now, graph both inequalities. For x > 5, use a dashed line at x = 5 and shade to the right. For x < 1, use a dashed line at x = 1 and shade to the left:

    |          Shaded Region (|x - 3| > 2)    
    |          
    |          
    |          
    |          
    |          
----|----
    1          5
    |          

Summary

• Graph linear inequalities using dashed or solid lines based on the type of inequality.
• Graph compound inequalities by shading the region between the lines.
• For quadratic inequalities, find the roots and test intervals to determine where the inequality holds.
• For absolute value inequalities, rewrite them as two separate inequalities.

Check your understanding

1. How would you graph the inequality 3x - 5 > 1?
2. What type of line would you use for the inequality x + 4 <= 6?
3. For the inequality x^2 > 9, what intervals will you test?
4. How do you graph |x + 2| < 3?