BellworkBellwork1.What is a solution to an equation or

inequality?

2.Determine whether 2/3 is a solution to this equation: 3x - 10 = -8.

3. Determine if -1/2 is a solution to 4x – 3 < -12 ?

Solving Linear Solving Linear Equations and Equations and

InequalitiesInequalities

Section 1.3/1.6

VocabularyVocabulary Equation: a mathematical

sentence that contains an equal sign.

Inequality: a mathematical sentence that contains an inequality symbol.

Solution: a number that makes the sentence true.

To solve an To solve an Equation/Inequality:Equation/Inequality:

Isolate the variable!

Don't forget to flip the inequality symbol if you multiply or divide by a negative

Always write the variable on the left when finished with the problem.

Find X!Find X!

3 Types of Solutions to 3 Types of Solutions to EquationsEquations

1. The equation has ONE REAL solution

(which will be x = #)

2. The equation has ANY REAL NUMBER as a solution

(which will be 0 = 0)

3. The equation has NO REAL solution

(which will be 0 = #)

1. 3/7 x + 9 = 15

- 9 - 9

3/7x = 6

7/3(3/7x) = 6 (7/3)

x = 14

2. 5n + 11 = 7n - 9

- 11 - 11

5n = 7n -20

-7n -7n

-2n = -20 -2 -2 n = 10

Solve each, then check your solutionSolve each, then check your solution

4(3x - 5) = -2(-x + 8) - 6x

12x - 20 = 2x - 16 - 6x

12x - 20 = - 4x - 16 + 20 + 20 12x = -4x + 4 +4x +4x 16x = 4 16 16 x = 1/4

4. 1/3x + 1/4 = X - 1/6

12(1/3x + 1/4 = X - 1/6)

4X + 3 = 12X - 2

- 3 -3

4X = 12X - 5

-12X -12X

-8X = -5

-8 -8

X = 5/8

Solve each, then check your solutionSolve each, then check your solution 3.

5. -4(6x - 5) = 23 - 3(8x + 1)

-24x + 20 = 23 – 24x – 3

-24x + 20 = 20 – 24x

- 20 -20

-24x = -24x

+ 24x +24x

0 = 0

All real numbers

6. -2x - (x + 3) = 3x – 3

-2x – x – 3 = 3x – 3

-3x – 3 = 3x – 3

+ 3 + 3

-3x = 3x + 0

-3x -3x

-6x = 0

-6 -6

X = 0

Solve each, then check your solutionSolve each, then check your solution

Solve each, then check your solutionSolve each, then check your solution

7. -2/3(6/5x - 7/10) = 17/20

-12/15x + 7/15 = 17/20

60(-12/15x + 7/15 = 17/20)

-48x + 28 = 51

-28 -28

-48x = 23

-48 -48

X = - 23/48

8. 2(x - 7) + 3 = 2x – 14

2X – 14 + 3 = 2x –14

2X – 11 = 2x – 14

+ 11 + 11

2X = 2x – 3

-2x -2x

0 = -3

No Solution!

Without solving...Without solving...

Tell whether the equation below has no solution, one solution, or any real solution, then explain.

a) 5x + 2 = 5x + 2 _____________

b) 5x + 2 = 2 – 5x _____________

c) 3x – 2 = 3x _____________

d) 2(x – 7) + 3 = 2x + 14 _____________

e) 3(2x – 1) = 6x – 3 _____________

One, X = #

No Sol, 0 = #

No Sol, 0 = #

ARN, 0=0

ARN, 0 = 0

Types of Solutions to Types of Solutions to InequalitiesInequalities

Set Notation Interval Notation

Graph

X < # (-∞ , # )

X < # (-∞ , # ]

X > # (#,∞)

X > # (#,∞ ]

Solve and graph each on a number line. Put Solve and graph each on a number line. Put

the answers in interval notation.the answers in interval notation. 1.  5y – 8 < 12 5Y < 20 Y < 4

(-∞, 4)

 

                         

2.  2x + 1 < 6x – 1 -4x < -2 X > ½

[ ½,∞)

Solve and graph each inequality.  Write the solution in Solve and graph each inequality.  Write the solution in interval notation. interval notation.

3.  3 < 2/3 x + ½ 18 < 4x + 3 15 < 4x 15/4 < x X > 15/4 ( 15/4, ∞)

4.  7x + 9 > 10x – 12 -3x > -21 X < 7 (-∞, 7]

Solve and graph each inequality.  Write the solution in Solve and graph each inequality.  Write the solution in interval notation. interval notation.

5.  1/3(5/8x + 2) > 2/3 x + 2 5/24x + 2/3 > 2/3x + 2 24[5/24x + 2/3 > 2/3x + 2] 5x + 16 > 16x + 48 -11X > 32 x < -32/11 ( -∞, -32/11]

4.  4(1/2x + 7/3) > 1/3(x + 12) 2x + 28/3 > 1/3x + 4 3[2x + 28/3 > 1/3x + 4] 6x + 28 > x + 12 5X > -16 X > -16/5 (-16/5, ∞)

Word Problems!Word Problems!

Mack works at an appliance store. He earns a base salary of $16,000 plus 12% of the total value of the appliances he sells for the year (commission). Last year he earned $18,800. What was the total value of the appliances he sold?

Coach Barr has decided to put a walking track of uniform width around the outside of the football field and then fence in the entire area. The football field is 160 ft by 300 ft and he has 968 ft of fencing to put around the entire area (football field AND walking track). What is the maximum width of the walking track?

1. A real estate broker's salary is $18,000. She earns a 4% commission on total sales. How much must she sell to earn $55,000 total.

2. You plan to frame a 5 in by 7 in photo. The frame will have a uniform width. The perimeter of the framed photo will be 4o in. What is the width of the frame?

3. A restaurant server earns a base salary of $6 per hour plus tips. If he averages $12 per hour in tips, how many hours must he work to ear a total of $333?

4. The formula to convert Fahrenheit to Celsius is F = 9/5 C + 32. If the temperature outside is 45 degrees Fahrenheit, what is the temperature in Celsius?

5. A garden is 3 feet by 5 feet. If you want to put a sidewalk around the garden that is uniform width, what is the width of the sidewalk if the perimeter of the garden and sidewalk is 28 feet ?