P-3

Linear Equations and Inequalities

Vocabulary Linear EquationLinear Equation in in oneone variable. variable.

AAx + x + BB = = CC

AA ≠ 0≠ 0BB and and CC are constants are constants

You’ve seen this before!You’ve seen this before!

44x x –– 22 == 66

Give me another example of a Give me another example of a linear equation linear equation in one variable.in one variable.

+ 2 +2+ 2 +2

4x = 8 4x = 8 ÷ 4 ÷4÷ 4 ÷4

x = 2 x = 2

Linear Equation in 2 Variables

AAx + x + BBy = y = CCLinear Equation in 3 Linear Equation in 3 VariablesVariables

33x + x + 44y = y = 1212

AAx + x + BBy + y + CCz = z = DD

33x + x + 44y + y + 66z = z = 1212

There is no limit to the number of variables in There is no limit to the number of variables in a linear equation. Animation uses over 100.a linear equation. Animation uses over 100.

What makes it a ‘linear” equation ?

If the exponent of the variable(s) is a ‘1’, then it is a linear equation.

3x + 4y + 6z = 12

Solutions of Linear equationsWhat does “What does “solutionsolution” mean ?” mean ?

Vocabulary Vocabulary SolutionSolution: the number the variable must equal: the number the variable must equal

in order to make the statement true.in order to make the statement true.

x + 1 = 2x + 1 = 2 The The solutionsolution is: x = 1 is: x = 1

“Solving” Linear equationsWhat does it mean to “What does it mean to “solvesolve” an equation?” an equation?

Vocabulary Vocabulary SolveSolve: using properties to re-write the equation: using properties to re-write the equation

in the form: x = (some exact value) in the form: x = (some exact value) in order to make the statement true.in order to make the statement true.

x + 1 = 2x + 1 = 2

(subtraction property of equality)(subtraction property of equality) -1 -1-1 -1

x = 1x = 1 (solution)(solution)

Distributive Property of Addition over Multiplication

2(x + 4) = (2 * x) + (2 * 4) = 2x + 8

Biggest 2 errors in the distributive property:

Trying to multiply when the operation is add or subtract

Failing to distribute a negative to BOTH terms inside parentheses

5 - (x - 4) = 5 x – 20 NO NO NO!!!

= 5 – x – 85 - (x - 4) NO NO NO!!!

Your turn: Solve these equations.

1. 1. xx 4372

2. 2.

3. 3.

317

5

x

3

2

4

3x

4. 4. 5325

32

x

x

VocabularyLinear Inequality (in one variable): Ax + B < C or Ax + B > C or Ax + B ≤ C or Ax + B ≥ C.

3x – 2 < 4

More Vocabulary

Equivalent InequalitiesEquivalent Inequalities: : An inequality that has the same solution as the original inequality. .

x + 2 = 4

x = 2

x + 2 < 4

-2-2 -2-2

-2-2-2-2x < 2

(subtraction property of inequality)

(subtraction property of equality)

Equivalent EquationsEquivalent Equations: : An equation that has the same solution as the original equation.

SolvingSolving inequalities inequalities (variable on both sides of a single inequality symbol)

3x + 1 ≤ 2x + 6

KEY POINT: collect variable on the side that will result in a positive coefficient.

-2x -2x

x + 1 ≤ 6

-1 -1

x x ≤ 5≤ 5

Your turn: Solve the inequality

7. 7. -14x – 2 < 5x + 6

5. 5. 2x + 2 ≤ 6

6. 6. 2(x – 3 ) ≥ 8

The “Gotcha” of Inequalities

2 – 2x ≤ 6

+ 2x + 2x

2 ≤ 2x + 6

-6 -6

-4 ≤ 2x

÷2 ÷2

-2 ≤ x

2 – 2x ≤ 6

-2 -2

-2x ≤ 4

÷ -2 ÷ -2

x ≤ -2

Anytime you multiply or divide by a negative number, you must switch the direction of the inequality !!

SolvingSolving inequalities inequalities (variable on both sides of a single inequality symbol)

3x + 1 ≤ 2x + 6

To avoid the “gotcha”: collect the variable on the side that will result in a positive coefficient.

-2x -2x

x + 1 ≤ 6

-1 -1

x x ≤ 5≤ 5

Your turn: Solve the inequality

10. 2(x – 4) 10. 2(x – 4) < 4x + 6< 4x + 6

8. 2x – 6 8. 2x – 6 ≤ 3 – x ≤ 3 – x

9. 18 + 2x 9. 18 + 2x ≥ 9x + 4≥ 9x + 4

Compound inequalities Compound inequalities (two inequality symbols)

5 ≤ x + 1 and

-1 -1

4 4 ≤ x≤ x

Same as: 4 ≤ x < 8Same as: 4 ≤ x < 8

x + 1 < 9

and

-1 -1

x x < 8< 8

5 ≤ x + 1 < 9

Compound inequalities Compound inequalities (two inequality symbols)

5 ≤ x + 1 < 9KEY POINT: subtraction property of inequality do the same thing (left-middle-right)-1 -1 -1

4 4 ≤ x < 8≤ x < 8

Same as: 4 ≤ x 4 ≤ x andand x < 8 x < 8

Your turn: Solve the inequality

11. 11. -3 < 4 – x ≤ 3

12. 12. -5 < x + 1 and x + 1 ≤ 6

Solving inequalities Solving inequalities (“or” type)

x - 2 ≤ 3 or x + 2 > 8

KEY POINT: treat “or” type compound inequalities as two separate inequalities.

+2 +2

x x ≤ 5≤ 5

-2 -2-2 -2

x > x > 66or

x x ≤ 5≤ 5 x > x > 66or

Your turn: Solve the inequality

13. 13. 4x - 7 ≤ 5 or 3x + 2 > 23

14. 14. x + 1 ≤ -3 or x – 2 > 0

Sometimes there is no solution

2(x – 4) 2(x – 4) > 2x + 1 > 2x + 1

SolutionSolution: the value(s) of the variable that make: the value(s) of the variable that make the statement the statement true.true.

2x – 8 2x – 8 > 2x + 1 > 2x + 1 -2x -2x -2x -2x

– – 8 8 > 1 > 1

No solutionNo solution: when the : when the variable dissappears variable dissappears and the and the resulting resulting statement is falsestatement is false..

Sometimes the solution is all real numbers.

Solution: the value(s) of the variable that make the statement true.

4x – 5 ≤ 4(x + 2)

-4x -4x

– 5 ≤ 8

Infinitely many solutionsInfinitely many solutions: : when the variable dissappears and the resulting statement is true.

4x – 5 ≤ 4x + 8

Your turn: Solve the inequality

15. 15. 2(3x – 1) > 3(2x + 3)

16. 16. 2x + 3 ≤ 3(x + 2) – x

Graphing Single Variable inequalitiesGraphing Single Variable inequalities

x > 3

1 2 3 4 5

What part of the number line is greater than 3 ?What part of the number line is greater than 3 ?

Graphing Single Variable inequalitiesGraphing Single Variable inequalities

x < 5

1 2 3 4 5

What part of the number line is less than 5 ?What part of the number line is less than 5 ?

Your turn: Graph the following

1717. x ≥ 7

18. 18. 3 > x

Graphing Compound InequalitiesGraphing Compound Inequalities

x > 3 and x < 5

1 2 3 4 5

What part is What part is x > 3x > 3 ? ?

And means both conditions must be met

What part is What part is x < 5x < 5??

What is the What is the intersectionintersection or or overlapoverlap of the two? of the two?

Vocabularyx > 3 and x < 5

HintHint: Inequality with “and” looks like: : Inequality with “and” looks like:

Compound inequalityCompound inequality

1 2 3 4 5

HintHint: This can also be written as: 3 < x < 5: This can also be written as: 3 < x < 5

Your turn: Graph the following compound inequalities.

x > 2 and x < 619. 19.

20. 20. -2 < x ≤ 5

Graphing “Graphing “oror” type compound inequalities.” type compound inequalities.

x ≤ 3 or x > 5OrOr means: the points that satisfy means: the points that satisfy eithereither condition condition

1 2 3 4 5

Which part is x > 5 ?Which part is x ≤ 3 ?

HintHint: inequality with “OR” looks like: : inequality with “OR” looks like:

Your turn:2121. Solve and graph the compound inequality: :

1 2 3 4 5

2x + 3 ≤ 5 or x - 3 > 2

Verbal InequalitiesThe cost of a car is at most $20,000.

It takes Jehah It takes Jehah no less thanno less than 5 minutes to run a mile. 5 minutes to run a mile.

It takes It takes betweenbetween 3 and 8 months to build a house. 3 and 8 months to build a house.

The cost of a loaf of bread is The cost of a loaf of bread is less than less than $2$2

You You can’t buy can’t buy a car for a car for less than less than $8000.$8000.

Your turn: Your turn: (a) Write in inequality notation (b) Graph the inequality

It never gets above 100 degrees in Huntsville.

22. 22. There are least 65,000 spectators at the game.

23. 23.

24. 24. You can fit, at most, 5 cars in your garage.

Three Ways to show an Inequality

1. Inequality: x > 3

2. Bracket Notation: (3, )

3. Number line Notation:

x ≤ 2

1 2 3 4 5 6

( , 2]

Inequalities Involving Fractions

3

1

42

1

3xx

Another example?

3

1

2

2

8

5

xx

Your Turn: Solve this inequality

3

1

5

13

2

32

xxx25. 25.

Homework

P-3: evens: 2-10, 18-26, 32-44, 54

(18 problems)