7.5

Linear Inequalities

7.5 – Linear Inequalities

Goals / “I can…”Graph linear inequalitiesWrite and use linear inequalities when modeling

real – world situations

7.5 – Linear Inequalities

RECALLRECALLGraph x > 4 on a number line.

Graph y ≤ 2 on a number line.

0 2 4-2-4

0 2 4-2-4

7.5 – Linear Inequalities

REMEMBER:The type of dot on the number line is

important. OpenOpen DOT means NOT INCLUDED (> or <> or <)ClosedClosed DOT means INCLUDED (≤ or ≥≤ or ≥)

Remember these symbols!!!!

Less than

Greater than

Less than or EQUAL TO

Greater than or EQUAL TO

7.5 – Linear Inequalities

When graphing inequalities on the coordinate plane, we use a similar idea. Dashed linesDashed lines mean the same as open circles.

(> or <> or <)

Solid lines meanSolid lines mean the same as closed circles. (≤ or ≥≤ or ≥)

Graphing an Inequality in Two Variables

Graph x < 2Step 1: Start by graphing the line x = 2

Now what points would give you less

than 2?

Since it has to be x x << 2 2 we shade everything to

the left of the line.

7.5 – Linear Inequalities

Oh, and did I mention we have to shade a part of the graph?????

When considering shading, you shade the part of the graph that WORKS FOR THE WORKS FOR THE EQUATIONEQUATION.

Graphing a Linear Inequality

Sketch a graph of y 3

7.5 – Linear Inequalities

Graph

x ≥ 3x ≥ 3

7.5 – Linear Inequalities

Graph

y ≤ 2x + 2y ≤ 2x + 2

Solving an Inequality

Solving a linear inequality in one variable is much like solving a linear equation in one variable. Isolate the variable on one side using inverse operations.

Add the same number to EACH side.

x – 3 < 5Solve using addition:

53 x+3 +3

x < 8

Solving Using Subtraction

Subtract the same number from EACH side.

106 x

4x-6 -6

THE TRAP…..

When you multiply or divide each side of an inequality by a negative negative numbernumber, you must REVERSE the inequality SYMBOL to maintain a true statement.

Solving by multiplication of a negative #

Multiply each side by the same negative number and REVERSE the inequality symbol.

4 x Multiply by (-1).

4x

(-1) (-1)

See the switch

Solving by dividing by a negative #

Divide each side by the same negative number and reverse the inequality symbol.

62 x

3x

-2 -2

Solving using Multiplication

Multiply each side by the same positive number.

32

1x

6x

(2) (2)

Some Helpful Hints•If the sign is > or < the line is

dashed

•If the sign is or the line will be solid

When dealing with just x and y.

•If the sign > or the shading either goes up or to the right

•If the sign is < or the shading either goes down or to the left

Using What We KnowSketch a graph of x + y < 3

Step 1: Put into slope intercept form

y < -x + 3

Step 2: Graph the line y = -x + 3

When dealing with slanted lines

•If it is > or then you shade above

•If it is < or then you shade below the line