Solving and Graphing Linear Inequalities

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Linear Inequalities

Math 10 – Ms. Albarico

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Students are expected to:

•Express and interpret constraints using inequalities.

•Graph equations and inequalities and analyse graphs, both with andwithout graphing technology.

•Investigate and make and test conjectures about the solution toequations and inequalities using graphing technology.

•Solve problems using graphing technology.

•Write an inequality to describe its graph.

•Solve linear and simple radical, exponential, and absolute valueequations and linear inequalities.

•Interpret solutions to equations based on context.

•Relate sets of numbers to solutions of inequalities.

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Key Terms

•Inequality

•Constraints

•Feasible Region

•System of Equation

•Optimal Solution

•Objective Function

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•constraint – a restriction on the allowable values of a variable in aproblem

•feasible points – data points that satisfy the constraints given

•feasible region – a shaded region on a graph indicating that all pointswithin the region are possible solutions to the problem

• feasible solution – any solution to a problem that is possible withinthe constraints

• inequality – a mathematical statement that shows that two numericalor variable expressions are not always equal

• graphical solutions – solutions obtained by graphing the feasibleregion.

•intersection point – the point where two graphs cross each other andwhere both graphs are equal

•objective function – a function that allows you to find the maximumor minimum values using given constraints

•optimal solution – the solution that best meets the constraints in theproblem

•system of equations – two or more equations involving the samevariable quantities

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What’s an Inequality?

•Is a range of values,

rather than ONE set number

• An algebraic relation showing that aquantity is greater than or less thananother quantity.

Speed limit:

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Inequality SymbolsInequality Symbols

Lessthan

Greater than

Lessthan orequal to

Greater thanor equal to

Notequal to

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Solutions….

You can have a range of answers……

-5 -4 -3 -2 -1 0 1 2 3 4 5

All real numbers less than 2

x< 2

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Solutions continued…

-5 -4 -3 -2 -1 0 1 2 3 4 5

All real numbers greater than -2

x > -2

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Solutions continued….

-5 -4 -3 -2 -1 0 1 2 3 4 5

All real numbers less than or equal to 1

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Solutions continued…

-5 -4 -3 -2 -1 0 1 2 3 4 5

All real numbers greater than or equal to -3

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Did you notice,

Some of the dots were solidand some were open?

-5 -4 -3 -2 -1 0 1 2 3 4 5

Why do you think that is?

If the symbol is > or < then dot is open because it can not beequal.

If the symbol is  or  then the dot is solid, because it can bethat point too.

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Write and Graph a LinearInequality

Sue ran a 2-K race in 8 minutes. Write an inequalityto describe the average speeds of runners who werefaster than Sue. Graph the inequality.

Faster average speed >

Distance

Sue’s Time

-5 -4 -3 -2 -1 0 1 2 3 4 5

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Solving an Inequality

Solving a linear inequality in one variable is much likesolving a linear equation in one variable. Isolate thevariable on one side using inverse operations.

Add the same number to EACH side.

x – 3 < 5

Solve using addition:

+3 +3

x < 8

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Solving Using Subtraction

Subtract the same number from EACH side.

-6 -6

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Using Subtraction…

Graph the solution.

-5 -4 -3 -2 -1 0 1 2 3 4 5

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Using Addition…

-5 -4 -3 -2 -1 0 1 2 3 4 5

Graph the solution.

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THE TRAP…..

When you multiply or divide each side ofan inequality by a negative number, youmust reverse the inequality symbol tomaintain a true statement.

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Solving using Multiplication

Multiply each side by the same positive number.

(2)

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Solving Using Division

Divide each side by the same positive number.

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Solving by multiplication of anegative #

Multiply each side by the same negative numberand REVERSE the inequality symbol.

Multiply by (-1).

(-1)

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See the switch

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Solving by dividing by anegative #

Divide each side by the same negativenumber and reverse the inequality symbol.

-2

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Linear Inequality

•Inequality with one variable to the firstpower.

for example:

2x-3<8

•A solution is a value of the variable thatmakes the inequality true.

x could equal -3, 0, 1, etc.

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Transformations for InequalitiesTransformations for Inequalities

•Add/subtract the same number on eachside of an inequality

•Multiply/divide by the same positivenumber on each side of an inequality

•If you multiply or divide by a negativenumber, you MUST flip the inequalitysign!

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Ex: Solve the inequality.

2x-3<8

+3 +3

2x<11

2 2

Flip the sign after dividingby the -3!

Graphing Linear Inequalities

•Remember:

< and > signs will have an open dot o

and signs will have a closed dot

graph of graph of

-3

-2

-1

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Example: Solve and graph the solution.

Compound Inequality

•An inequality joined by “and” or “or”.

Examples

“and”“or”

think between

think oars on a boat

-4 -3 -2 -1 0 12

-3 -2 -1 0 1 2 3 45

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Example: Solve & graph.

-9 < t+4 < 10

-13 < t < 6

Think between!

-13

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Last example! Solve & graph.

-6x+9 < 3 or -3x-8 > 13

-6x < -6 -3x > 21

x > 1 or x < -7

Flip signs

Think oars

-7

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Graphing a Linear Inequality

Graphing a linear inequality isvery similar to graphing a linearequation.

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Graphing Inequalities

The solution is the set of all points in theregion that is common to all the inequalitiesin that system. It is the region where theshadings overlap. This solution is called thefeasible region.