INTRODUCTION TO LINEAR INEQUALITIES

What is linear inequalities and how do we solve it?

Answer: When solving linear inequalities, we use the same concepts that we use when solving linear equations. Basically we still want to get the variable on one side and everything else on the other side by using inverse operations. The difference is, when a variable is equal to one number, that number is the only solution, like x=5. But, when a variable is greater than or less than a number, there are infinite number of values that would be a part of the answer, like x>5. x>5 means that X( the variable) is everything greater then 5. So If you were asked to graph x>5 you would do what is shown in example one. When you see a < that means greater then or equal to. That means if you saw x< 2, then x is everything less then or equal to 2. Look at example 2 for example.

Example 1:

x>5

beg_alg_tut18ex1a.gif

Since the equation says that x is greater than 5, 5 is not one of the answer, therefore, use an open dot. You want to show on the number line what X can be. This example shows X is everything greater then 5.

Example 2.
x < 2

beg_alg_tut18ex2a.gif

Since the equation indicates that x is less than or equal to 2, we close the dot in the middle to indicate that 2 can also be the answer.

How to simplify a Linear inequality?


Inequalities can be solved or rearranged using the same rules you have used for equations with one exception. Multiplying or dividing both sides of an inequality by a negative number reverses the relationship. For example: Given: -8x<44
You would wan to multiply by -1/8(or divide by -8). So you have to reverse the inequality sign when you multiply or divide both sides of an inequality. Note that < becomes >. x>-5.5

How To graph a Inequality?


When you graph inequalities, the inequality sign in the equation will determine whether the line is solid or dashed and whether the area above of below the line is shaded. For example, y≥2x+2 will be a solid line with the area above it shaded, meaning that the points on the line as well as the shaded area can complete the equation.

However if the equation was y<2x+2 you would graph it with a dashed line and shade below the line. This means the points to complete the equation are just below the line. Look at example 3

Example 3.
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