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Module 4: Equations & Inequalities
After completion of this unit, you will be able to…
Learning Target #1: Creating and Solving Linear Equations
• Solve one, two, and multi-step equations (variables on both sides)
• Justify the steps for solving a linear equation
• Create and solve an equation from a context
Learning Target #2: Creating and Solving Linear Inequalities
• Solve and graph a linear inequality
• Create and solve an inequality from a context
• Name and graph a compound inequality
• Give solutions for a compound inequality
Learning Target #3: Isolating a Variable
• Solve a literal equation (multiple variables) for a specified variable
• Use a Formula to Solve Problems
Timeline for Unit 4
|Monday |Tuesday |Wednesday |Thursday |Friday |
|October 1st |2nd |3rd |4th |5th |
|Day 1 – |Day 2 – |Day 3 – |Day 4 - | |
|Solving 1 & 2 Step Equations, |Multi-Step Equations |Multi-Step Equations, |Equations with Fractions & |Mixed Practice Day |
|Intro to Algeblocks | |Properties of Equality |Decimals | |
|8th |9th |10th |11th |12th |
|Day 5 – |Day 6 – Creating & Solving |PSAT Day |Early Release |Quiz over Days 1-5 |
|Creating & Solving Equations |Equations/Activity | |Extra Practice on Solving | |
|from a Context | | |Equations | |
|15th |16th |17th |18th |19th |
|Day 7 – |Day 8 – |Day 9 – |Review, |Day 10 – |
|Solving Inequalities |Compound Inequalities |Creating & Solving Inequalities|Quiz over Days 6-9 |Solving Literal Equations |
| | |from a Context | | |
|22nd |23rd |24th |25th |26th |
|Day 11 – |Day 12 – |Day 13 – | | |
|Solving Literal Equations |Review Day |Unit 4 Assessment |Midterm Review |MIDTERM |
|(Complex) | | | | |
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Day 1 – Solving One & Two Step Equations
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Remember, an expression is a mathematical “phrase” composed of terms, coefficients, and variables that stands for a single number, such as 3x + 1 or x2 – 1. We use Properties of Operations to simplify algebraic expressions. Expressions do NOT contain equal signs.
An equation is a mathematical “sentence” that says two expressions are equal to each other such as 3x + 1 = 5. We use Properties of Equality (inverse operations) to solve algebraic equations. Equations contain equal signs.
When solving equations, you must perform inverse operations, which means you have to perform the operation opposite of what you see. You must also remember the operation you perform on one side of the equation must be performed to the other side.
|Informal |Formal |
|Operation |Inverse |Property |General Example | Example 1 |
|Addition | |Addition Property of Equality |If a = b, |If x – 4 = 8, then x = 12 |
| | | |then a + c = b + c | |
|Subtraction | |Subtraction Property of |If a = b, |If x + 5 = 7, then x = 2 |
| | |Equality |then a – c = b - c | |
|Multiplication | |Multiplication Property of |If a = b, |If [pic], then x = 18 |
| | |Equality |then ac = bc | |
|Division | |Division Property of Equality |If a = b, |If 2x = 10, then x = 5 |
| | | |then [pic] | |
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No More “Cancelling”
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When you first learned to solve equations in middle school, you might have used the words “cancel”. We are no longer going to use the word “cancel”. Take a look at the following examples:
( Adding the opposite
Additive inverse ( Multiplying by the Reciprocal
Adding to zero Multiplicative Inverse
Divides/Multiplies to one
|Additive Inverse |A number plus its inverse equals 0. |a + -a = 0 |7 + -7 = 0 |
|Multiplicative Inverse |A number times its reciprocal equals|a ∙ [pic]= 1 |3 ∙ [pic]= 1 |
|(Reciprocal) |1. | | |
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Solving One Step Equations Practice
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Practice: Solve each equation.
1. x – 4 = 3 Operation You See: _______________ Inverse Operation: _______________
2. y + 4 = 3 Operation You See: _______________ Inverse Operation: _______________
3. [pic] Operation You See: _______________ Inverse Operation: _______________
4. 6p = 12 Operation You See: _______________ Inverse Operation: _______________
Practice: Solve each equation on your own.
a. x – 6 = 10 b. -5d = 25 c. 8 + m = -4
d. [pic] e. y – (-9) = 2 f. [pic]
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Solving Two Step Equations
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When solving equations with more than one step, you still want to think about how you can “undo” the operations you see. For the following equations, describe the operations being performed on each variable (go in PEMDAS order). Then describe the inverses using a table.
a. 3x + 5 = 14 b. 2n – 6 = 4 c. [pic]
[pic]
Practice: Solve each equation, showing all steps, for each variable.
1. 3x - 4 = 14 2. 2x + 4 = 10 3. 7 – 3y = 22
4. 0.5m – 1 = 8 5. -6 + [pic] = -5 6. [pic]
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Error Analysis with Solving Equations
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1. William solved the following equation on his homework last night. However, he solved it incorrectly. Describe the mistake William made and what he should have done instead. Then re-solve the equation to find the correct answer.
Mistake: _______________________________________________________________
________________________________________________________________________
________________________________________________________________________
Corrected Solution:
2. Tyler solved the following equation on his homework last night. However, he solved it incorrectly. Describe the mistake Tyler made and what he should have done instead. Then re-solve the equation to find the correct answer.
Mistake: _______________________________________________________________ ________________________________________________________________________
________________________________________________________________________
Corrected Solution:
[pic]Day 2 – Solving Multi-Step Equations
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Multi-step equations mean you might have to add, subtract, multiply, or divide all in one problem to isolate the variable. When solving multi-step equations, you are using inverse operations, which is like doing PEMDAS in reverse order.
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Multi - Step Equations with Combining Like Terms
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Practice: Solve each equation, showing all steps, for each variable.
a. -5n + 6n + 15 – 3n = -3 b. 3x + 12x – 20 = 25 c. -2x + 4x – 12 = 40
[pic]
Multi - Step Equations with the Distributive Property
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Practice: Solve each equation, showing all steps, for each variable.
a. 2(n + 5) = -2 b. 4(2x – 7) + 5 = -39 c. 6x – (3x + 8) = 16
[pic]
Multi – Step Equations with Variables on Both Sides
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Practice: Solve each equation, showing all steps, for each variable
a. 5p – 14 = 8p + 4 b. 8x – 1 = 23 – 4x c. 5x + 34 = -2(1 – 7x)
[pic]
Error Analysis with Solving Equations
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1. Rachel solved the following equation on her homework. However, she solved it incorrectly. Describe the mistake Rachel made and what she should have done instead. Then resolve the equation to find the correct answer.
Mistake: ___________________________________________________________
___________________________________________________________________
____________________________________________________________________
Correction Solution:
2. Mikayla solved the following equation on her homework. However, she solved it incorrectly. Describe the mistake Mikayla made and what she should have done instead. Then resolve the equation to find the correct answer.
Mistake: ____________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Correction Solution:
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Day 3 –Justifying the Solving of Equations
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|Properties of Addition Operations |
|Property |What It Means |General Example | Example 1 |
|Commutative Property of Addition |Rearrange the order and the sum will|a + b = b + a |2 + 4 = 4 + 2 |
| |stay the same. | | |
|Associative Property of Addition |Change the order of the grouping and|(a + b) + c = a + (b + c) |(4 + 6) + 1 = 4 + (6 + 1) |
| |the sum will stay the same. | | |
|Additive Identity |Zero added to any number will equal |a + 0 = a |4 + 0 = 4 |
| |that number. | | |
|Additive Inverse |A number plus its inverse equals 0. |a + -a = 0 |7 + -7 = 0 |
|Properties of Multiplication Operations |
|Commutative Property of Multiplication |Rearrange the order and the product |a ∙ b = b ∙ a |5 ∙ 2 = 2 ∙ 5 |
| |will stay the same. | | |
|Associative Property of Multiplication |Change the order of the grouping and|(a ∙ b) ∙ c = a ∙ (b ∙ c) |(3 ∙ 4) ∙ 2 = 3 ∙ (4 ∙ 2) |
| |the product will stay the same. | | |
|Multiplicative Identity |One times any number equals that |a ∙ 1 = a |8 ∙ 1 = 8 |
| |number. | | |
|Multiplicative Inverse |A number times its reciprocal equals|a ∙ [pic]= 1 |3 ∙ [pic]= 1 |
|(Reciprocal) |1. | | |
|Zero Property of Multiplication |Any number times 0 will always equal|a ∙ 0 = 0 |7 ∙ 0 = 0 |
| |0. | | |
|Distributive Property |Multiply a number to every term |a(b + c) = ab + ac |4(x + 5) = 4x + 4(5) |
| |within a quantity (parenthesis). | |= 4x + 20 |
Practice: Each of the following expressions has been simplified one step at a time. Next to each step, identify the property or simplification used in the step.
1. 4 + 5(x + 7) Given 2. 4(10x + 2) – 40x Given
4 + (5x + 35) _______________________________ 40x + 8 – 40x _______________________________
5x + 4 + 35 _______________________________ 8 + 40x – 40x _______________________________
5x + (4 + 35) _______________________________ 8 + 0 _______________________________
5x + 39 _______________________________ 8 _______________________________
[pic]
Justifying the Solving of Equations
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|Properties of Equality |
|Property |General Example | Example 1 |
|Addition Property |If a = b, |If x – 4 = 8, then x = 12 |
| |then a + c = b + c | |
|Subtraction Property |If a = b, |If x + 5 = 7, then x = 2 |
| |then a – c = b - c | |
|Multiplication Property |If a = b, |If [pic], then x = 18 |
| |then ac = bc | |
|Division Property |If a = b, |If 2x = 10, then x = 5 |
| |then [pic] | |
|Reflexive Property |a = a |5 = 5 |
|Symmetric Property |If a = b, then b = a |If 2 = x, then x = 2 |
|Transitive Property |If a = b and b = c, then a = c |If x + 2 = y and y = 4x + 3, |
| | |then x + 2 = 4x + 3 |
|Substitution Property |If x = y, then y can be substituted for x in any |If x = 3 and the expression is 2x – 7, then 2(3) - 7 |
| |expression | |
Practice: Using properties of operations and equality, list each property next to each step in the equation solving process.
Example 1
|x + 4 = 9 |Given |
|x = 5 | |
|7 = x - 5 |Given |
|12 = x | |
|x = 12 | |
Example 2
Example 3
|[pic] = 5 |Given |
|x = 15 | |
Example 4
|6x = 24 |Given |
|x = 4 | |
[pic]
Justifying the Solutions to Two & Multi-Step Equations
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Practice: Identify the property or simplification that is used in each step to solve the equation.
Example 1
|3x + 5 = -13 |Given |
|3x = -18 | |
|x = -6 | |
Example 2
|12 = 2(x – 4) |Given |
|12 = 2x – 8 | |
|20 = 2x | |
|10 = x | |
|x = 10 | |
Example 3
|5n – 3 = 2(n + 3) + 9 |Given |
|5n – 3 = 2n + 6 + 9 | |
|5n – 3 = 2n + 15 | |
|3n – 3 = 15 | |
|3n = 18 | |
|n = 6 | |
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Special Types of Solutions
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Solve the following equations. What do you notice about the solutions?
a. 2x – 7 + 3x = 4x + 2 b. 3(x – 5) + 11 = x + 2(x + 5) c. 3x + 7 = 5x + 2(3 – x) + 1
We are going to use the graphing calculators to view what these equations look like. Draw a sketch of the graphs:
Problem A: Problem B: Problem C:
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Conclusions:
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Day 4 – Equations with Fractions and Decimals
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When you solve equations with decimals, you solve them as if you would an equation without decimals.
1. 3.5x – 37.9 = .2x 2. 14.7 + 2.3x = 4.06 3. -1.6 – 0.9w = 11.6 + 2.4w
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Equations with Fractions
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When solving equations with fractions, you want to find a way to eliminate the fraction.
When You Could Multiply by the Reciprocal (One Fraction Attached to the Variable):
1. [pic] 2. [pic] 3. [pic] 4. [pic]
When You Should Multiply by a Common Denominator (Two Fractions)
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic] 6. [pic]
7. [pic] 8. [pic] 9. [pic]
When You Could Cross Multiply (Can also be solved by multiplying by a common denominator):
1. [pic] 2. [pic] 3. [pic]
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Day 5 – Creating Equations from a Context
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Explore: Read the scenario below and answer the following questions.
Annie is throwing a graduation party. She wants to send nice invitations to all of her guests. She found a company that will send her a pack of 10 personalized invitations for $6 each, plus a $5 shipping fee for the entire order, no matter how many invitations she orders.
a. What is the total cost of Annie purchasing three packs of invitations?
b. What is the total cost of Annie purchasing five packs of invitations?
c. What is the total cost of Annie purchasing ten packs of invitations?
d. Describe how you calculated the cost of each order.
e. Write an algebraic expression that represents the total cost of any order. Let p represent the number of invitation packs that were ordered.
f. How many packs of invitations were ordered if the total cost of the order was $53?
g. How many packs of invitation were ordered if the total cost of the order was $29?
h. Describe how you calculated the number of invitation packs ordered for any order amount.
i. Write an equation to describe this situation. Let p represent the number of invitation packs ordered and c represent the total cost of the order.
j. Use your equation to determine how many invitation packs Annie ordered if her total cost was $47.
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Earlier in our unit, you learned to write expressions involving mathematical operations. You used the following table to help you decode those written expressions. We are going to use those same key words along with words that indicate an expression will become part of an equation or inequality.
|Addition |Subtraction |Multiplication |Division |Equals |
|Sum |Difference |Of |Quotient |Is |
|Increased by |Decreased by |Product |Ratio of |Equals |
|More than |Minus |Times |Percent |Will be |
|Combined |Less |Multiplied by |Fraction of |Gives |
|Together |Less than |Double |Out of |Yields |
|Total of |Fewer than |Twice |Per |Costs |
|Added to |Withdraws |Triple |Divided by | |
|Gained | | | | |
|Raised | | | | |
|Plus | | | | |
When taking a word problem and translating it to an equation or inequality, it is important to “talk to the text” or underline/highlight key phrases or words. By doing this it helps you see what is occurring in the problem.
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Modeling Mathematics with Equations
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A person’s maximum heart rate is the highest rate, in beats per minute that the person’s heart should reach. One method to estimate the maximum heart rate states your age added to your maximum heart rate is 220. Using this method, write and solve an equation to find the maximum heart rate of a 15 year old.
In the equation above, we did not know one of the quantities. When we do not know one of the quantities, we use a variable to represent the unknown quantity. When creating equations, it is important that whatever variable you use to represent the unknown quantity, you define or state what the variable represents.
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Practice Examples: In the examples below, “talk to the text” as you translate your word problems into equations. Define a variable to represent an unknown quantity, create your equation, and then solve your equation.
1. Six less than four times a number is 18. What is the number?
Variables: ___________________________
Equation: ___________________________
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2. You and three friends divide the proceeds of a garage sale equally. The garage sale earned $412. How much money did each friend receive?
Variables: ___________________________
Equation: ___________________________
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3. On her iPod, Mia has rock songs and dance songs. She currently has 14 rock songs. She has 48 songs in all. How many dance songs does she have?
Variables: ___________________________
Equation: ___________________________
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4. Brianna has saved $600 to buy a new TV. If the TV she wants costs $1800 and she saves $20 a week, how many months will it take her to buy the TV (4 weeks = 1 month)?
Variables: ___________________________
Equation: ___________________________
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5. It costs Raquel $5 in tolls to drive to work and back each day, plus she uses 3 gallons of gas. It costs her a total of $15.50 to drive to work and back each day. How much per gallon is Raquel paying for her gas?
Variables: ___________________________
Equation: ___________________________
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6. Mrs. Jackson earned a $500 bonus for signing a one year contract to work as a nurse. Her salary is $22 per hour. If her first week’s check including the bonus is $1204, how many hours did Mrs. Jackson work?
Variables: ___________________________
Equation: ___________________________
[pic]
7. Morgan subscribes to a website for processing her digital pictures. The subscription is $5.95 per month and 4 by 6 inch prints are $0.19. How many prints did Morgan purchase if the charge for January was $15.83?
Variables: ___________________________
Equation: ___________________________
[pic]
8. A rectangle is 12m longer than it is wide. Its perimeter is 68m. Find its length and width (Hint: p = 2w + 2l).
Variables: ___________________________
Equation: ___________________________
[pic]
9. The daycare center charges $120 for one week of care. Families with multiple children pay $95 for each additional child per week. Write an equation for the total cost for one week of care in terms of the number of children. How many children does a family have it they spend $405 a week in childcare?
Variables: ___________________________
Equation: ___________________________
[pic]
10. The party store has a special on greeting cards. It charges $14 for 4 greeting cards and $1.50 for each additional card. Write an equation for the total cost of greeting cards in terms of the number of cards. What is the total cost for 9 greeting cards?
Variables: ___________________________
Equation: ___________________________
[pic]
11. Clara has a coupon for $10 off her favorite clothing store. The coupon is applied before any discounts are taken. The store is having a sale and offering 15% off everything. If Clara has $50 to spend, how much can her purchases total before applying the discount to her coupon?
Variables: ___________________________
Equation: ___________________________
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Day 6 – Creating Equations from a Context (Complex)
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|Consecutive Numbers Chart |
|Type of Consecutive Numbers |Examples |Expressions for Terms |
| | |First Second Third |
|Consecutive Numbers |4, 5, 6 |x |x + 1 |x + 2 |
| |27, 28, 29 | | | |
|Consecutive Even Numbers |8, 10, 12 |x |x + 2 |x + 4 |
| |62, 64, 66 | | | |
|Consecutive Odd Numbers |23, 25, 27 |x |x + 2 |x + 4 |
| |89, 91, 93 | | | |
1. The sum of three consecutive numbers is 72. What is the smallest of these numbers?
Variables: ____________________________________
Equation: ____________________________________
2. Find three consecutive odd integers whose sum is 261.
Variables: ____________________________________
Equation: ____________________________________
[pic]
Equations Based Off Multiple Scenarios
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Scenario: Five friends have a certain number of DVDs:
• Colby has the fewest.
• Jace has 7 more than Colby.
• Jessie has twice as many as Jace.
• Isaac has 3 times as many as Colby.
• Amara has 6 less than Jace.
If the friends have a total of 182 DVDs together, then how many does each person have?
[pic]
Day 7 – Solving Inequalities
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An inequality is a statement that that compares two quantities. The quantities being compared use one of the following signs:
[pic]
When reading an inequality, you always to want to read from the variable. Translate the inequalities into words:
A. x > 2 ___________________________________________________________________________
B. -3 > p ___________________________________________________________________________
C. y ≤ 0 ___________________________________________________________________________
D. -2 ≤ z ___________________________________________________________________________
E. x ≠ 1 ___________________________________________________________________________
[pic]
When graphing an inequality on a number line, you must pay attention to the sign of the inequality.
|Words |Example |Circle |Number Line |
|Greater Than |x > 2 |Open |[pic] |
|Less Than |p < -3 |Open |[pic] |
|Greater Than or Equal To |z ≥ -2 |Closed |[pic] |
|Less Than or Equal To |y ≤ 0 |Closed |[pic] |
|Not Equal To |x ≠ 1 |Open |[pic] |
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Solutions to Inequalities
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A solution to an inequality is any number that makes the inequality true.
| | | |
|Value of x |2x – 4 ≥ -12 |Is the inequality true? |
| | | |
|-2 | | |
| | | |
| | | |
|-4 | | |
| | | |
| | | |
|-6 | | |
[pic]
Solving and Graphing Linear Inequalities
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Solving linear inequalities is very similar to solving equations, but there is one minor difference. See if you can figure it out below:
Conclusion:
Practice: Solve each inequality and graph.
1. x - 4 < -2 2. -3x > 12
[pic] [pic]
3. 7 ≤ ½x 4. [pic]
5. -2(x + 1) ≥ 6 6. 6x – 5 ≤ 7 + 2x
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Day 8 - Compound Inequalities
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Scenario: is an online store that offers discounts on sports equipment to high school athletes. When customers buy items from the side, they must pay the cost of the items as well as a shipping fee. At , a shipping fee is added to each order based on the total cost of all the items purchased. The table below provides the shipping fee categories for .
[pic]
Using the inequality symbols you learned yesterday, fill in the boxes to represent each shipping fee category if x represents the total cost of items purchased.
Compound Inequalities Compact Form
a. $6.50 shipping fees: x $0.01 and x $20
b. $9.00 shipping fee x $20 and x $50
// c. $11.00 shipping fee x $50 and x $75
// d. $12.25 shipping fee x $75 and x $100
/
e. $13.10 shipping fee x $100
/
When two simple inequalities are combined into one statement using the words OR or AND, the result is a compound inequality. Compound inequalities that use the word AND can be written in compact form. Compound inequalities that use the word OR cannot be put into compact form; the word OR must be used.
/
Difference between AND & OR:
/
/
/
[pic]
Writing Compound Inequalities from a Graph[pic]
Practice: Identify each of the numbers as an AND or OR compound inequality. Then write the compound inequality.
a. b.
[pic] [pic]
/ /
Possible Solutions: Possible Solutions:
c. d.
[pic]/ / [pic]
Possible Solutions: Possible Solutions:
[pic]
Writing Compound Inequalities from a Context[pic]
Practice: Define a variable for each unknown quantity and write a compound inequality for each scenario.
1. Water becomes non-liquid when it is 32[pic]F or below, or when it is at least 212[pic]F. Write an inequality to describe water in its non-liquid form.
2. Every day, a female needs to eat at least 1500 calories, but less than 1800 calories. Write an inequality to describe the amount of calories a female should have per day.
3. Each type of fish thrives in a specific range of temperatures. The optimum temperatures for sharks range from 18 degrees Celsius to 22 degrees Celsius. Write an inequality that represents the temperatures where sharks will NOT thrive.
[pic]
Day 9 – Creating Inequalities from a Context[pic]
When creating problems that involve inequalities, you will use the same methods as creating equations, except you have new keywords that will replace the equal sign with an inequality sign.
|< |≤ |> |≥ |
|Less than |Less than or equal to |Greater than |Greater than or equal to |
|Fewer than |At most |More than |At least |
| |Maximum | |Minimum |
| |No more than | |No less than |
Examples: Define a variable for the unknown quantity, create an inequality, and then solve.
1. One half of a number decreased by 3 is no more than 33.
Variables: ____________________________________
Inequality: ____________________________________
2. Alexis is saving to buy a laptop that costs $1,100. So far she has saved $400. She makes $12 an hour babysitting. What’s the least number of hours she needs to work in order to reach her goal?
Variables: ____________________________________
Inequality: ____________________________________
3. Keith has $500 in a savings account at the bank at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 each week for food, clothes, and movie tickets. How many weeks can Keith withdraw money from his account?
Variables: ____________________________________
Inequality: ____________________________________
4. Yellow Cab Taxi charges a $1.75 flat rate in addition to $0.65 per mile. Katie has no more than $10 to spend on a ride. How many miles can Katie travel without exceeding her limit?
Variables: ____________________________________
Inequality: ____________________________________
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Day 10 – Isolating a Variable
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Isolating a variable simply means to solve for that variable or get the variable “by itself” on one side of the equal sign (usually on the left). Sometimes we may have more than one variable in our equations; these type of equations are called literal equations. We solve literal equations the same way we solve “regular” equations.
Steps for Isolating Variables
1. Locate the variable you are trying to isolate.
2. Follow the rules for solving equations to get that variable by itself.
|Solving an Equation You’re Familiar with |Solving a Literal Equation |
|2x = 10 | gh = m solve for h |
| | |
| | |
| | |
| | |
|2x + 5 = 11 | ax + b = c solve for x |
| | |
| | |
| | |
| | |
| | |
Practice:
1. Solve the equation for b: a = bh 2. Solve the equation for b: y = mx + b
3. Solve the equation for x: 2x + 4y = 10 4. Solve the equation for m: y = mx + b
5. Solve the equation for w: p = 2l + 2w 6. Solve the equation for a: [pic]
Your Turn:
7. Solve the equation for y: 6x – 3y = 15 8. Solve the equation for h: [pic]
[pic]
1. You are visiting a foreign county over the weekend. The forecast is predicted to be 30 degrees Celsius. Are you going to pack warm or cold clothes? Use Celsius = [pic].
2. The area of a triangle is given by the formula A = ½bh or[pic], where b is the base and h is the height.
a. Use the formula given to find the height of the triangle that has a base of 5 cm and an area of 50 cm.
b. Solve the formula for the height.
c. Use the formula from part b to find the height of a triangle that has a base of 5 cm and an area of 50 cm.
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Day 12 – Isolating a Variable (Complex)
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One of the most important skills you will encounter for the next two units is the ability to take an equation in standard form (Ax + By = C) and solve for y. You had a few problems from yesterday like this, but take some time to practice a few more.
a. 5x – 2y = 8 b. -3x + 3y = 6 c. -7x – 4y = 12
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Solving Literal Equations with Distribution
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When solving problems with distribution or common factors, you do NOT want to distribute the term on the outside. Instead divide both sides by the factor on the outside of the parenthesis.
Note: If you see you are trying to solve for a variable in multiple locations, move all the terms with that variable to one side and then factor that variable out.
a. Solve a(y + 1) = b for y. b. 3ax – b = d – 4cx for x
c. 4x – 3m = 2mx – 5 for x. d. 4x + b = 2x + c for x.
[pic]
Complex Literal Equations
[pic]
a. [pic] for a b. [pic]for b c. [pic] for W
d. p(t + 1) = -2, for t e. [pic] for x f. [pic] for A
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Ways to Eliminate Fractions
• Multiply by the Reciprocal
• Multiply by a Common Denominators – always works!
• Cross Multiply
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Age
Added to
Maximum Heart Rate
Is
220
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Experiment
Take the inequality 6 > 2. Is this true?
1. Add 3 to both sides. What is your new inequality? 2. "#UVWXY“ÕÖ 1 p “ ” Á éÒ¸Ò¸¤?kXF5F5F5F5F5 hÄs?hÄSubtract 3 from both sides. What is your new inequality?
3. Multiply both sides by 3. What is your new inequality? 4. Divide both sides by 3. What is your new inequality?
3. Multiply both sides by -3. What is your new inequality? 4. Divide both sides by -3. What is your new inequality?
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Think About It…
1. What is the least amount a customer can spend on items and pay $6.50 for shipping? __________________
2. What is the greatest amount a customer can spend on items and pay $6.50 for shipping? ________________
3. What is the shipping fee if Michael spends $75? Why?
Compound Inequalities in Graph Form:
[pic]
Smallest number ≤ x ≤ largest number
[pic]
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