4.1 Solve and Graph Linear Inequalities

Morty Morterson; carl123

4.1 Solve and Graph Linear Inequalities

When given an equation, such as $x = 4$ or $x = -5,$ there are specific values for the variable. However, with inequalities, there is a range of values for the variable rather than a defined value. To write the inequality, use the following notation and symbols:

Symbol	Meaning
	> Greater than
	≤ Greater than or equal to
	< Less than
	≥ Less than or equal to

Example 4.1.1

Given a variable $x$ such that $x > 4$ , this means that $x$ can be as close to 4 as possible but always larger. For $x > 4$ , $x$ can equal 5, 6, 7, 199. Even $x =$ 4.000000000000001 is true, since $x$ is larger than 4, so all of these are solutions to the inequality. The line graph of this inequality is shown below:

x > 4

Written in interval notation, $x > 4$ is shown as $(4, \infty)$ .

Example 4.1.2

Likewise, if $x < 3$ , then $x$ can be any value less than 3, such as 2, 1, −102, even 2.99999999999. The line graph of this inequality is shown below:

x < 3

Written in interval notation, $x < 3$ is shown as $(-\infty, 3)$ .

Example 4.1.3

For greater than or equal (≥) and less than or equal (≤), the inequality starts at a defined number and then grows larger or smaller. For $x \ge 4,$ $x$ can equal 5, 6, 7, 199, or 4. The line graph of this inequality is shown below:

x ≥ 4

Written in interval notation, $x \ge 4$ is shown as $[4, \infty)$ .

Example 4.1.4

If $x \le 3$ , then $x$ can be any value less than or equal to 3, such as 2, 1, −102, or 3. The line graph of this inequality is shown below:

x ≤ 3

Written in interval notation, $x \le 3$ is shown as $(-\infty, 3].$

When solving inequalities, the direction of the inequality sign (called the sense) can flip over. The sense will flip under two conditions:

First, the sense flips when the inequality is divided or multiplied by a negative. For instance, in reducing $-3x < 12$ , it is necessary to divide both sides by −3. This leaves $x > -4.$

Second, the sense will flip over if the entire equation is flipped over. For instance, $x > 2$ , when flipped over, would look like $2 < x.$ In both cases, the 2 must be shown to be smaller than the $x$ , or the $x$ is always greater than 2, no matter which side each term is on.