Linear Inequalities

Inequality Signs

Read left to right: a < b    a is less than b

a < b   a is less than or equal to b a > b    a is greater than b

a > b    a is greater than or equal to b

Graphing a Linear Inequality

Graphing a linear inequality is

very similar to graphing a linear equation.

Graphing a Linear Inequality

1) Solve the inequality for y

(or for x if there is no y).2) Change the inequality to an

equation

and graph.3) If the inequality is < or >, the line

is dotted. If the inequality is ≤ or

≥, the line is solid.

Graphing a Linear Inequality

4) To check that the shading is correct, pick a

point in the area and plug it into the inequality. 5) If the inequality statement is true, the shading is correct. If the inequality statement is false, the shading is

incorrect.

Example 1:  Graph x < 2. Since we needed to indicate all values less than or equal to 2,  the part of the number line that is to the left of 2 was darkened. Since there is an equal line under the < symbol, this means we do include the endpoint 2.  We can notate that by using a closed hole (or you can use a boxed end).

Example 2:  Graph x > 5

Since we needed to indicate all values greater than 5,  the part of the number line that is to the right of 5 was darkened.

Since there is no equal line under the > symbol, this means we do not include the endpoint 5 itself.  We can notate that by using an open hole (or you can use a curved end).

Addition/Subtraction Property for Inequalities

If a < b, then a + c < b + c If a < b, then a - c < b - c

Example 3:   

Solve the inequality and graph the solution set.

Multiplication/Division Properties for Inequalities

If a < b  AND  c is positive, then   ac < bc If a < b  AND  c is positive, then   a/c < b/c

In other words, multiplying or dividing the same POSITIVE number to both sides of an inequality does not change the inequality.

when multiplying/dividing by a positive value

Example 5:  Solve the inequality and graph the solution set. 

Example 6:  Solve the inequality and graph the solution set. 

Multiplication/Division Properties for Inequalities

If a < b  AND  c is negative, then   ac > bc If a < b  AND  c is negative, then   a/c > b/c

when multiplying/dividing by a negative value

The reason for this is, when you multiply or divide an expression by a negative number, it changes the sign of that expression.  On the number line, the positive values go in a reverse or opposite direction than the negative numbers go, so when we take the opposite of an expression,  we need to reverse our inequality to indicate this.

Example 7:  Solve the inequality and graph the solution

I multiplied by a -2 to take care of both the negative and the division by 2 in one step.  In line 2, note that when I did show the step of multiplying both sides by a -2, I reversed my inequality sign.

Strategy for Solving a Linear Inequality

Step 1:  Simplify each side, if needed.  

This would involve things like removing ( ), removing fractions, adding like terms, etc.Step 2:  Use Add./Sub. Properties to move the variable term on one side and all other terms to the other side.  

Step 3: Use Mult./Div. Properties to remove any values that are in front of the variable.

Example 10:  Solve the inequality and graph the solution

Even though we had a -2 on the right side in line 5, we were dividing both sides by a positive 2, so we did not change the inequality sign. 

Example 11:  Solve the inequality and graph the solution

Graphing a Linear Inequality Pick a point, (1,2), in the shaded area. Substitute into the original inequality 3 – x > 0 3 – 1 > 0 2 > 0 True! The inequality has been graphed correctly.

6

4

2

3

Graphing Linear Inequalities: y > mx + b, etc

Graph the solution to y < 2x + 3.  Just as for number-line inequalities, the first step is to find

the "equals" part.  In this case, the "equals" part is the line y = 2x + 3. There are a couple ways you can graph this: you can use a T-chart, or you can graph from the y-intercept and the slope. Either way, you get a line that looks like this:

Graphing a Linear Inequality

Graph the inequality 3 - x > 0First, solve the inequality for x.

3 - x > 0

-x > -3

x < 3

Graph: x<3 Graph the line x = 3.

Because x < 3 and not x ≤ 3, the line will be dotted.

Now shade the side of the line where x < 3 (to the left of the line).

6

4

2

3

Now we're at the point where your book gets complicated, with talk of "test points" and such. When you did those one-variable inequalities (like x < 3), did you bother with "test points", or did you just shade one side or the other? Ignore the "test point" stuff, and look at the original inequality:  y < 2x + 3.

You've already graphed the "or equal to" part (it's just the line); now you're ready to do the "y less than" part. In other words, this is where you need to shade one side of the line or the other. Now think about it: If you need y LESS THAN the line, do you want ABOVE the line, or BELOW? Naturally, you want below the line. So shade it in:

Solving linear inequalities

Graph the solution to 2x – 3y < 6.  

First, solve for y: 2x – 3y < 6

–3y < –2x + 6 y > ( 2/3 )x – 2

[Note the flipped inequality sign in the last line. Don't forget to flip the inequality if you multiply or divide through by a negative!

Now you need to find the "equals" part, which is the line y = ( 2/3 )x – 2. It looks like this:

By using a dashed line, you still know where the border is, but you also know that it isn't included in the solution.

Since this is a "y greater than" inequality, you want to shade above the line, so the solution looks like this: