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Unit 7: Equations & inequalities
About this unit.
Solving equations is a superpower. It means we can model a situation with an equation in any way that makes sense to us, even with an unknown value in the middle. Inequalities are for situations with many true options, like how many pages I can send in my letter using just 1 stamp.
Algebraic equations basics
- Variables, expressions, & equations (Opens a modal)
- Testing solutions to equations (Opens a modal)
- Intro to equations (Opens a modal)
- Testing solutions to equations Get 5 of 7 questions to level up!
One-step equations intuition
- Same thing to both sides of equations (Opens a modal)
- Representing a relationship with an equation (Opens a modal)
- Dividing both sides of an equation (Opens a modal)
- One-step equations intuition (Opens a modal)
- Identify equations from visual models (tape diagrams) Get 3 of 4 questions to level up!
- Identify equations from visual models (hanger diagrams) Get 3 of 4 questions to level up!
- Solve equations from visual models Get 3 of 4 questions to level up!
One-step addition & subtraction equations
- One-step addition & subtraction equations (Opens a modal)
- One-step addition equation (Opens a modal)
- One-step addition & subtraction equations: fractions & decimals (Opens a modal)
- One-step addition & subtraction equations Get 5 of 7 questions to level up!
- One-step addition & subtraction equations: fractions & decimals Get 5 of 7 questions to level up!
One-step multiplication and division equations
- One-step division equations (Opens a modal)
- One-step multiplication equations (Opens a modal)
- One-step multiplication & division equations (Opens a modal)
- One-step multiplication & division equations: fractions & decimals (Opens a modal)
- One-step multiplication equations: fractional coefficients (Opens a modal)
- One-step multiplication & division equations Get 5 of 7 questions to level up!
- One-step multiplication & division equations: fractions & decimals Get 5 of 7 questions to level up!
Finding mistakes in one-step equations
- Finding mistakes in one-step equations (Opens a modal)
- Find the mistake in one-step equations Get 3 of 4 questions to level up!
One-step equation word problems
- Modeling with one-step equations (Opens a modal)
- Translate one-step equations and solve Get 3 of 4 questions to level up!
- Model with one-step equations Get 3 of 4 questions to level up!
- Model with one-step equations and solve Get 3 of 4 questions to level up!
Intro to inequalities with variables
- Testing solutions to inequalities (Opens a modal)
- Plotting inequalities (Opens a modal)
- Plotting an inequality example (Opens a modal)
- Inequalities word problems (Opens a modal)
- Graphing inequalities review (Opens a modal)
- Testing solutions to inequalities (basic) Get 3 of 4 questions to level up!
- Graphing basic inequalities Get 3 of 4 questions to level up!
- Inequality from graph Get 3 of 4 questions to level up!
- Plotting inequalities Get 3 of 4 questions to level up!
- Inequalities word problems Get 3 of 4 questions to level up!
Dependent and independent variables
- Dependent & independent variables (Opens a modal)
- Dependent & independent variables: graphing (Opens a modal)
- Dependent and independent variables review (Opens a modal)
- Independent versus dependent variables Get 3 of 4 questions to level up!
- Tables from equations with 2 variables Get 3 of 4 questions to level up!
- Match equations to coordinates on a graph Get 3 of 4 questions to level up!
Analyzing relationships between variables
- Writing equations for relationships between quantities (Opens a modal)
- Analyzing relationships between variables (Opens a modal)
- Equations and inequalities FAQ (Opens a modal)
- Relationships between quantities in equations Get 3 of 4 questions to level up!
- Analyze relationships between variables Get 3 of 4 questions to level up!
Be Prepared
18 x + 30 18 x + 30
3 − 2 n 3 − 2 n
2 x − 6 2 x − 6
0.045 0.045
w − 3 w − 3
0.35 x + 0.4 0.35 x + 0.4
2 c + 3 2 c + 3
0.042 0.042
15 15 is greater than x x .
15 < x 15 < x
2 5 x + 4 2 5 x + 4
− x + 4 − x + 4
a. >; b. >; c. >; d. >
m = 2 m = 2
a = 0 a = 0
u = 2 u = 2
x = 4 x = 4
p = −2 p = −2
q = −8 q = −8
y = − 17 5 y = − 17 5
z = 0 z = 0
identity; all real numbers
conditional equation; q = − 9 11 q = − 9 11
conditional equation; k = 201 14 k = 201 14
contradiction; no solution
x = 1 2 x = 1 2
x = −2 x = −2
x = 12 x = 12
u = −12 u = −12
n = 2 n = 2
m = −1 m = −1
r = 1 r = 1
s = −8 s = −8
n = 9 n = 9
d = 16 d = 16
He bought two notebooks.
He did seven crosswords puzzles.
−15 , −8 −15 , −8
−29 , 11 −29 , 11
−33 , −32 , −31 −33 , −32 , −31
−13 , −12 , −11 −13 , −12 , −11
−10 , −8 , −6 −10 , −8 , −6
The average cost was $5,000.
The median price was $19,300.
ⓐ 36 ⓑ $26 ⓒ 125 % 125 %
ⓐ 33 ⓑ $36 ⓐ 175 % 175 %
8.8 % 8.8 %
ⓐ $600 ⓑ $1,800
ⓐ $2,975 ⓑ $11,475
He will earn $2,500.
She earned $7,020.
The rate of simple interest was 6%.
The rate of simple interest was 5.5%.
He paid $17,590.
She deposited $9,600.
b = 2 A h b = 2 A h
h = 2 A b h = 2 A b
C = 5 9 ( F − 32 ) C = 5 9 ( F − 32 )
b = 2 A − B h h b = 2 A − B h h
t = A − P P r t = A − P P r
r = A − P P t r = A − P P t
y = 9 − 4 x 7 y = 9 − 4 x 7
y = 1 − 5 x 8 y = 1 − 5 x 8
The window’s height is 12 meters.
The length of the base is 6 feet.
The measures of the angles are 20°, 70°, and 90°.
The measures of the angles are 30°, 60°, and 90°.
The length of the leg is 8.
The length of the leg is 12.
The length is 39 inches and the width is 16 inches.
The length is 17 yards and the width is 26 yards.
The lengths of the sides of the triangle are 5, 11 and 12 inches.
The lengths of the sides of the triangle are 4, 7 and 9 feet.
The length of the swimming pool is 70 feet and the width is 30 feet.
The length of the garden is 90 yards and the width is 60 yards.
The ladder reaches 12 feet.
He should attach the lights 8 feet from the base of the mast.
Jess has 41 nickels and 18 quarters.
Elane has 22 nickels and 59 dimes.
Eric bought thirty-two 49-cent stamps and twelve 35-cent stamps.
Kailee bought twenty-six 49-cent stamps and ten 20-cent stamps.
84 adult tickets, 31 student tickets
615 children’s tickets and 195 adult tickets
Orlando mixed five pounds of cereal squares and 25 pounds of nuts.
Becca mixed 21 gallons of fruit punch and seven gallons of soda.
The speed of the local train is 48 mph and the speed of the express train is 60 mph.
Jeromy drove at a speed of 80 mph and his mother drove 60 mph.
Christopher’s speed was 50 mph and his parents’ speed was 40 mph.
Ashley’s parents drove 55 mph and Ashley drove 62 mph.
Pierre and Monique will be 429 miles apart in 3 hours.
Thanh and Nhat will be 330 miles apart in 2.2 hours.
Suzy’s speed uphill is 1.8 1.8 mph and downhill is three mph.
The boat’s speed upstream is eight mph and downstream is12 mph.
Hamilton drove 40 mph in the city and 70 mph in the desert.
Phuong rode uphill at a speed of 12 mph and on the flat street at 20 mph.
Angie can buy 7 packs of juice.
Daniel can have 11 people at the party.
Sergio and Lizeth can travel no more than 500 miles.
Rameen can use no more than 76 therms.
Caleb must work at least 96 hours.
Elliot must work at least 85 jobs.
Brenda must babysit at least 27 hours.
Josue must shovel at least 20 driveways.
The homeowner can use 5–20 hcf and still fall within the “conservation usage” billing range.
The homeowner can use 16–40 hcf and still fall within the “normal usage” billing range.
ⓐ ± 2 ± 2 ⓑ no solution ⓒ 0
ⓐ ± 11 ± 11 ⓑ no solution ⓒ 0
x = 4 , x = − 2 3 x = 4 , x = − 2 3
x = −1 , x = 5 2 x = −1 , x = 5 2
x = 8 , x = 0 x = 8 , x = 0
x = 8 , x = 2 x = 8 , x = 2
No solution
x = − 2 5 , x = − 2 5 , x = 5 2 x = 5 2
x = 3 , x = 3 , x = 1 9 x = 1 9
The diameter of the rod can be between 79.991 and 80.009 mm.
The diameter of the rod can be between 74.95 and 75.05 mm.
Section 2.1 Exercises
y = 5 y = 5
w = −18 w = −18
q = 3 q = 3
x = 14 x = 14
c = 4 c = 4
c = 5 2 c = 5 2
y = −5 y = −5
s = 10 s = 10
p = −4 p = −4
c = −4 c = −4
h = 3 4 h = 3 4
m = 6 m = 6
conditional equation; j = 2 5 j = 2 5
conditional equation; m = 16 5 m = 16 5
x = −1 x = −1
y = −1 y = −1
a = 3 4 a = 3 4
w = 9 4 w = 9 4
b = 12 b = 12
p = −41 p = −41
x = − 5 2 x = − 5 2
n = −3 n = −3
u = 3 u = 3
x = 18 x = 18
x = 20 x = 20
d = 8 d = 8
L = 19.75 L = 19.75 feet
Answers will vary.
Section 2.2 Exercises
58 hardback books
−11 , −12 , −13 −11 , −12 , −13
−69 , −71 , −73 −69 , −71 , −73
ⓐ 54 ⓑ 108 ⓐ 30%
ⓐ 162.5 162.5 ⓑ $35 ⓐ 150%
$ 11.88 $ 11.88
−2.5 % −2.5 %
−11 % −11 %
ⓐ $ 26.97 $ 26.97 ⓑ $ 17.98 $ 17.98
ⓐ $576 ⓑ 30%
ⓐ $ 7.20 $ 7.20 ⓑ $ 23.20 $ 23.20
ⓐ $ 0.20 $ 0.20 ⓑ $ 0.80 $ 0.80
3.75 % 3.75 %
21.2 % 21.2 %
Section 2.3 Exercises
d = C π d = C π
L = V W H L = V W H
d 1 = 2 A d 2 d 1 = 2 A d 2
b 1 = 2 A h − b 2 b 1 = 2 A h − b 2
a = 2 h − 108 t t 2 a = 2 h − 108 t t 2
a = 180 − b − c a = 180 − b − c
p = 2 A − 2 B l p = 2 A − 2 B l
L = P − 2 W 2 L = P − 2 W 2
y = 15 − 8 x y = 15 − 8 x
y = −6 + 4 x y = −6 + 4 x
y = 4 + x y = 4 + x
y = 7 − 4 x 3 y = 7 − 4 x 3
y = 12 − 2 x 3 y = 12 − 2 x 3
y = 18 − 3 x −2 y = 18 − 3 x −2
45 ° , 45 ° , 90 ° 45 ° , 45 ° , 90 °
30 ° , 60 ° , 90 ° 30 ° , 60 ° , 90 °
18 meters, 11 meters
13.5 13.5 m, 12.8 12.8 m
25 ft, 50 ft
12 ft, 13 ft, 14 ft
3 ft, 6 ft, 8 ft
120 yd, 160 yd
40 ft, 85 ft
104 ° F 104 ° F
ⓐ Answers will vary. ⓑ The areas are the same. The 2 × 8 2 × 8 rectangle has a larger perimeter than the 4 × 4 4 × 4 square. ⓒ Answers will vary.
Section 2.4 Exercises
nine nickels, 16 dimes
ten $10 bills, five $5 bills
63 dimes, 20 quarters
16 nickels, 12 dimes, seven quarters
330 day passes, 367 tournament passes
40 postcards, 100 stamps
15 $10 shares, five $12 shares
34 general, 61 youth
114 general, 246 student
Four pounds of macadamia nuts, eight pounds almonds
3.6 3.6 lbs Bermuda seed, 5.4 5.4 lbs Fescue seed
$33,000 in Fund A, $22,000 in Fund B
5.9 % 5.9 %
Kathy 5 mph, Cheryl 3 mph
commercial 550 mph, private plane 340 mph
Violet 65 mph, Charlie 55 mph
Ethan 22 mph, Leo 16 mph
DaMarcus 16 mph, Fabian 22 mph
uphill 1.6 1.6 mph, downhill 4.8 4.8 mph
light traffic 54 mph, heavy traffic 30 mph
freeway 67 mph, mountain road 22.3 mph
running eight mph, walking three mph
ⓐ 15 minutes ⓑ 20 minutes ⓒ one hour (d) 1:25
Section 2.5 Exercises
A maximum of 14 people can safely ride in the elevator.
five drinks
260 messages
62 necklaces
seven lawns
Section 2.6 Exercises
5 ≤ n ≤ 24 5 ≤ n ≤ 24
6 ≤ w ≤ 12 6 ≤ w ≤ 12
ⓐ answers vary ⓑ answers vary
Section 2.7 Exercises
ⓐ x = 4 , x = −4 x = 4 , x = −4 ⓑ no solution ⓒ z = 0 z = 0
ⓐ x = 3 , x = −3 x = 3 , x = −3 ⓑ no solution ⓒ z = 0 z = 0
x = 1 , x = − 1 2 x = 1 , x = − 1 2
x = −1 , x = − 5 2 x = −1 , x = − 5 2
x = 7 , x = 1 x = 7 , x = 1
x = 1 , x = −5 x = 1 , x = −5
no solution
x = −1 , x = − 2 3 x = −1 , x = − 2 3
x = −3 , x = 3 x = −3 , x = 3
x = 2 , x = 1 4 x = 2 , x = 1 4
x = 3 , x = 2 x = 3 , x = 2
x = 3 , x = − 11 3 x = 3 , x = − 11 3
x = 3 2 , x = − 1 2 x = 3 2 , x = − 1 2
The minimum to maximum expected production is 207,500 to 2,225,000 bottles
The acceptable weight is 22.5 to 25.5 ounces.
Review Exercises
s = −22 s = −22
m = −14 m = −14
q = 18 q = 18
k = 3 4 k = 3 4
k = 23 k = 23
x = 5 x = 5
There are 116 people.
−3 , −10 −3 , −10
$ 3.89 $ 3.89
ⓐ $105 ⓑ 52.5 % 52.5 %
d 2 = 2 A d 1 d 2 = 2 A d 1
y = 4 x 3 − 4 y = 4 x 3 − 4
22.5 ° , 67.5 ° , 90 ° 22.5 ° , 67.5 ° , 90 °
24.5 24.5 cm, 12.5 12.5 cm
9 ft, 14 ft, 12 ft
nine pennies, six dimes, 12 quarters
57 students, 68 general admission
2.2 2.2 lbs of raisins, 7.8 7.8 lbs of nuts
9.7 % 9.7 %
Louellen 65 mph, Tracy 66 mph
upstream 3 mph, downstream 5 mph
heavy traffic 32 mph, light traffic 68 mph
$33 per day
at least $300,000
at least 112 jobs
x = 2 , x = 2 3 x = 2 , x = 2 3
x = 9 , x = −3 x = 9 , x = −3
The minimum to maximum expected usage is 210,000 to 220,000 bottles
Practice Test
x = −5 x = −5
a = 41 a = 41
x = 6 x = 6
x = −2 , x = − 1 3 x = −2 , x = − 1 3
−57 , −55 −57 , −55
12 dimes, seven quarters
2.5 2.5 hours
At most $55.56 per costume.
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- Authors: Lynn Marecek, Andrea Honeycutt Mathis
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- Book title: Intermediate Algebra 2e
- Publication date: May 6, 2020
- Location: Houston, Texas
- Book URL: https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
- Section URL: https://openstax.org/books/intermediate-algebra-2e/pages/chapter-2
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11 10. 8x+ 11 = 2(4x-7) + 25 = 8x -14+25 O Gina Wilson (All Things Algebra', LLC), 2012-2016 . Name: Date: Unit 2: Equations & Inequalities ... Unit 2: Equations & Inequalities Homework 13: Inequalities Review ** This is a 2-page document! ** Directions: Solve, graph, and write the solution to each inequality in interval notation. ...
We develop general methods for solving linear equations using properties of equality and inverse operations. Thorough review is given to review of equation solving from Common Core 8th Grade Math. Solutions to equations and inequalities are defined in terms of making statements true. This theme is emphasized throughout the unit.
Introduction to Systems of Equations and Inequalities; 7.1 Systems of Linear Equations: Two Variables; 7.2 Systems of Linear Equations: Three Variables; 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables; 7.4 Partial Fractions; 7.5 Matrices and Matrix Operations; 7.6 Solving Systems with Gaussian Elimination; 7.7 Solving Systems with Inverses; 7.8 Solving Systems with Cramer's Rule
Question: Name: Date: Unit 2: Equations & Inequalitles Homework 10: Multh-Step Inequalities Bell:_ Directions: Solve and express the fol lowing inequaities in interval L. 3 interval Notation: Interval Notation: 3.5+8)-7s 23 sx + 33 느 23 3373 Interval Notation: Interval Notation: 6. 2x +5 s 3x- 10 Interval Notation: Interval Notation: 7. 2(x-3)+ 5x s9x-14 Interval
Unit test. Level up on all the skills in this unit and collect up to 2,200 Mastery points! Start Unit test. Solving equations is a superpower. It means we can model a situation with an equation in any way that makes sense to us, even with an unknown value in the middle. Inequalities are for situations with many true options, like how many pages ...
Terms in this set (25) Study with Quizlet and memorize flashcards containing terms like Solve the inequality and graph on a number line. −4x+ 5 ≥ −11, Solve the inequality and graph on a number line. 2.5−3x < −15.5, What is the solution to the equation 4x+7=20 ? and more.
Alg1.2 Linear Equations, Inequalities, and Systems. In this unit, students expand and deepen their prior understanding of expressions, equations, and inequalities. Students reason about equations, inequalities, and systems of equations and inequalities as ways to represent constraints, and they reason about the process of solving equations and ...
Terms in this set (16) Study with Quizlet and memorize flashcards containing terms like Absolute value, Solving absolute value equations, Solving absolute value equations: Case 1 and more.
Unit 2: Equations and Inequalities with variables on both sides. Term. 1 / 10. Expression. Click the card to flip 👆. Definition. 1 / 10. A mathematical phrase that contains operations, numbers, and/or variables. Has no equal sign.
Supplemental Two-step equations (WS 2.3B) 2.2.6 Stations Activity 2.2.7 Solving Two-Step Equations 2.2 Exit Slip Quiz - Solving One and Two Step Equations 2.3.2 Solving Equations that Contain Like Terms 2.3.3 Solving Equations with Variables on Both Sides 2.3.1 Exit Slip 2.3.4 Practice Solving Equations 2.3.6 How Many Solutions? 2.3.2 Exit Slip
Description. This Multi-Step Equations and Inequalities Unit Bundle contains guided notes, homework assignments, three quizzes, study guide, and a unit test that cover the following topics: • Multi-Step Equations (Variables on One Side) • Multi-Step Equations (Variables on Both Sides) • Special Cases: Infinite Solution & No Solution.
2.2 Solve Equations using the Division and Multiplication Properties of Equality; 2.3 Solve Equations with Variables and Constants on Both Sides; 2.4 Use a General Strategy to Solve Linear Equations; 2.5 Solve Equations with Fractions or Decimals; 2.6 Solve a Formula for a Specific Variable; 2.7 Solve Linear Inequalities
Unit 5 - Systems of Linear Equations and Inequalities. This unit begins by ensuring that students understand that solutions to equations are points that make the equation true, while solutions to systems make all equations (or inequalities) true. Graphical and substitution methods for solving systems are reviewed before the development of the ...
For every real number a and b, and for c<0, if a>b, the ac<bc; if a<b, then ac>bc. Division Property of Inequality. Compound Inequality. Two inequalities that are joined by the word and or the word or. Study with Quizlet and memorize flashcards containing terms like solution of an inequality, equivalent inequalities, Addition Property of ...
For example, 4x2 and −9y2 don't have a common factor, but the whole polynomial is still factorable: 4x2−9y2 = (2x + 3y)(2x−3y). 3. Divide the x term into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression. 5. 7m. 7.
lanaskedelj. Preview. chapter 1. 11 terms. jaeden_micco. Preview. Study with Quizlet and memorize flashcards containing terms like For something to be considered an equation it MUST have ..., What is this considered 3x + 4, Is this an equation, expression, or inequality? 3x+6 > 10 and more.
2.1 Use a General Strategy to Solve Linear Equations; 2.2 Use a Problem Solving Strategy; 2.3 Solve a Formula for a Specific Variable; 2.4 Solve Mixture and Uniform Motion Applications; 2.5 Solve Linear Inequalities; 2.6 Solve Compound Inequalities; 2.7 Solve Absolute Value Inequalities
Unit 4 Equations and Inequalities; Unit 5 Systems of Equations and Inequalities; Unit 6 Quadratic Functions; Unit 7 Structures of Quadratic Expressions; Unit 8 More Functions, More Features; Unit 9 Modeling Data; Open Up HS Math is published as an Open Educational Resource.
Table of Contents for Common Core Algebra I. Unit 1 - The Building Blocks of Algebra. Unit 2 - Linear Expressions, Equations, and Inequalities. Unit 3 - Functions. Unit 4 - Linear Functions and Arithmetic Sequences. Unit 5 - Systems of Linear Equations and Inequalities. Unit 6 - Exponents, Exponents, Exponents and More Exponents.
multi-step equation. an equation that requires more than one step to solve. consecutive integers. integers in counting order. interval notation. Another way of expressing the solution set of an inequality using parentheses and brackets to show where the graph starts and ends; ex: [-2, + ∞) proportion. An equation stating that two ratios are ...
Now, we calculate the value of variable "x" by substituting the result of "y" into one of the equations. = −. = − =. Solution (10/9, 7/9). Independent System. 12. Find the solution of the following systems by elimination and determine if it is an independent, inconsistent or dependent system. + =. − − =.
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