# Algebra 1

Name: Date: Period:

Linear Equations and Inequalities in One Variable Linear Equation/Inequalities – equations/inequalities where the variable is raised to the first power and does not occur in a denominator, inside a square root symbol, or inside absolute value symbols.

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Page 349 #1 - 17 all Solve and Graph

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Page 349 #18 - 34 LEFT

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Page 349 #19 - 35 RIGHT – *****Quiz Tomorrow*****

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Page 356 #6 – 11

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Page 356 - 357 #22, 31, 34, 40, 55 and 58

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Page 356 – 357 #24, 33, 36, 45, 57 and 60 - **Quiz Tomorrow**

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Application Problems – Worksheet 1

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Application Problems – Worksheet 2

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AP - Page136 #48, Page 149 #55, Page 171 #53, Page 344 #39, Page 352 #14 (10) Chapter Review - *****Test Tomorrow***** (11) Page 194 #1 - 10, Page 388 #1 - 8

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Introduction – Properties (I, E) Properties will be used to justify your work throughout the unit. Properties of Addition Commutative Property of Addition The order in which two numbers are added does not change the sum a+b=b+a Example: 3 + (-2) = -2 + 3 Associative Property of Addition The way you group three numbers when adding does not change the sum (a + b) + c = a + (b + c) Example: (-5 + 6) + 2 = -5 + (6 + 2) Identity Property of Addition The sum of a number and 0 is the number a+0=a Example: -4 + 0 = -4 Property of Zero for Addition (aka Additive Inverse Property) The sum of a number and its opposite is 0. a + (-a) = 0 Example: 5 + (-5) = 0 Properties of Multiplication Commutative Property of Multiplication The order in which two numbers are multiplied does not change the product a*b=b*a Example: 3 * (-2) = -2 * 3 Associative Property of Multiplication The way you group three numbers when multiplying does not change the product (a * b) * c = a * (b * c) Example: (-6 * 2) * 3 = -6 * (2 * 3) Identity Property of Multiplication The product of a number and 1 is the number 1*a=a Example: -4 * 1 = -4 Property of Zero for Multiplication (aka Zero Product Property) The product of a number and 0 is 0. a*0=0 Example: (-2) * 0 = 0 Property of Opposites for Multiplication The product of a number and -1 is the opposite of the number. (-1) * a = -a Example: (-1)*(-3) = 3 Inverse Property of Multiplication (aka Multiplicative Inverse Property) The product of a number and its reciprocal is 1. 1 1 a* 𝑎 = 1 Example: 5 * 5 = 1 The Distributive Property The product of a and (b + c) a(b + c) = ab + ac (b + c)a = ba + ca The product of a and (b – c) a(b – c) = ab – ac (b – c)a = ba – ca 2|Page

Example: 5(x + 2) = 5x + 10 Example: (x + 4)8 = 8x + 32 Example: 4(x – 7) = 4x – 28 Example: (x – 5)9 = 9x – 45

Introduction – Properties (I, E) Properties of Equality Addition Property of Equality If a = b, then a + c = b + c

Example: If x = 3, then x + 2 = 5

Subtraction Property of Equality If a = b, then a – c = b – c

Example: if x = 7, then x – 6 = 1

Multiplication Property of Equality If a = b, then ca = cb

Example: if x = 12, then 3x = 36

Division Property of Equality 𝑎 𝑏 If a = b and c ≠ 0, then 𝑐 = 𝑐

Example: if 5x = 25, then x = 5 Examples

Directions: Identify the Algebraic property that supports each statement. 1) When three or more numbers are added, the sum is the same regardless of the grouping of the addends. For example (a + b) + c = a + (b + c) 2) Adding 0 to any number leaves it unchanged. For example a + 0 = a. 3) If you multiply the same number to both sides of an equation, the equation is still true. For example if a = b, then a x c = b x c. 4) The additive inverse of a number, a is -a so that a + -a = 0. 5) When two numbers are added, the sum is the same regardless of the order of the addends. For example a + b = b + a 6) If you divide the same number to both sides of an equation, the equation is still true. For example if a = b, then a / c = b / c. 7) The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example a x (b + c) = a x b + a x c 8) If you add the same number to both sides of an equation, the equation is still true. For example if a = b, then a + c = b + c. 9) If you subtract the same number from both sides of an equation, the equation is still true. For example if a = b, then a - c = b - c. 10) Multiplying any number by 0 yields 0. For example a x 0 = 0. 3|Page

6.3 Solving Compound Inequalities (E) Compound Inequality An inequality that consists of two inequalities connected by and or or. And – means the intersection of both sets or the elements that both sets have in common Or – means the union of both sets or combining all of the elements from both sets into one (E1)

Graph x < 4 and x ≥ 2

(E2)

Graph x < 2 or x ≥ 4

(E3)

Which inequalities describe the following graph?

A. B. C. D.

y > -3 or y < -1 y > -3 and y < -1 y ≤ -3 or y ≥ -1 y ≥ -3 and y ≤ -1

(E4)

Graph 6 < m < 8

(E5)

Which is equivalent to -3 < y < 5? A. B. C. D.

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y > -3 or y < 5 y > -3 and y < 5 y < -3 or y > 5 y < -3 and y > 5

6.3 Solving Compound Inequalities (E)

(P5)

Which is equivalent to x > -5 and x ≤ 1? A. B. C. D.

-5 < x ≤ 1 -5 > x ≥ 1 -5 > x ≤ 1 -5 < x ≥ 1

(E6)

Graph 2x < -6 or 3x ≥ 12

(E7)

Graph 3 < 2m – 1 < 9

(P7)

Graph x < 2 or x ≥ 4

(E8)

Graph x ≥ -1 or x ≤ 3

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6.4 Solving Absolute Value Equations and Inequalities (I, E) Absolute Value Equation/Inequality Any equation or inequality that contains an absolute value symbol is an absolute value equation/inequality. These equations/inequalities need to be split and solved separately. Special Cases 1. | | = negative Therefore ∅

2. | | < negative Therefore ∅

Steps: 1. Isolate the absolute value 2. Split the absolute value or =, >, ≥ 3. Solve each equation/inequality 4. Check your solutions 5. Graph your solutions

and negative Therefore ℝ

6.4 Solving Absolute Value Equations and Inequalities (I, E)

(E4)

Solve and Graph

|8 + 5a| = 14 – a

(E5)

Solve and Graph

|x + 6 |< 20

(P5)

Solve and Graph

|3x - 4 |≤ 8

(E6)

Solve and Graph

|½x + 7 |> 19

(E7)

Solve and Graph

-4|2x + 17 |≥ 12

(E8)

Solve and Graph

|x|> x - 1

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Linear Equations and Inequalities: Applications and Problem Solving (E) Problem Solving Strategy 1. 2. 3. 4. 5.

READ and RE-READ the problem UNDERLINE or HIGHLIGHT the main idea(s) DEFINE the variable (or the unknown) Use a VERBAL MODEL to help set-up the equation/inequality WRITE & SOLVE the equation/inequality

Example 1: WHEELCHAIRS…The van used to transport patients to and from a rehabilitation

facility is equipped with a wheelchair lift. The maximum lifting capacity for the lift is 300 pounds. The wheelchairs used by the facility weigh 55 pounds each. What is the maximum weight of a wheelchair occupant who can use the lift?

Example 2: DANCE CLASSES…The Central Bucks School Gymnastics charges $24 per class and a one-time registration fee of $15. A student paid a total of $687 to the school. Find the numbers of classes the student took.

Example 3: BASKETBALL…Stub Hub (a ticket agency) is selling tickets to the Philadelphia Sixers’s game. The agency charges $32.50 for each ticket, a convenience charge of $3.30 for each ticket, and a processing fee of $5.90 for the entire order. The total charge for an order is $220.70. How many tickets were purchased?

Example 4: HIGH SPEED INTERNET…Garrett and Brett are getting high-speed Internet access at the same time. Garret’s provider charges $60 for installation and $42.95 per month. Brett’s provider has free installation and charges $57.95 per month. After how many months will they both have paid the same amount for high-speed Internet service?

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Linear Equations and Inequalities: Applications and Problem Solving (E)

Example 5: LUGGAGE WEIGHT…You are checking a bag at the Philadelphia International Airport. Bags can weigh no more than 50 pounds. Your bag weighs 16.8 pounds. Find the possible weights w (in pounds) that you can add to the bag.

Example 6: PILOTING…A student pilot plans to spend 80 hours on flight training to earn a private license. The student has saved $6000 for training. Find and solve an inequality to find the possible hourly rates r that the student can afford to pay for training.

Example 7: CAR WASH…Use the sign below. A gas station charges $.10 less per gallon of gasoline if a customer also gets a car wash. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $20?

Example 8: TEMPERATURE… Under ideal conditions, water will remain a liquid when the temperature F (in degrees Fahrenheit) satisfies the inequality 32 < F < 212. Write the inequality in degrees Celsius. 9 F = 5C + 32

Example 9: BASKETBALLS…Before the start of a basketball game, a basketball must be inflated to an air pressure of 8 pounds per square inch (psi) with an absolute error of 0.5 psi. (Absolute error is the absolute deviation of a measured value from an accepted value.) Find the minimum and maximum acceptable air pressures for the basketball.

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