Ohio State University Solving Equations Questions

Foundations of Mathematical Reasoning Version 3.0 (2020)Student Pages
Preview Assignment 15.D
Preparing for the next class
In the next class, you will need to know how to apply the order of operations when
simplifying an expression, how to square a number, and how to solve a two-step linear
equation.
Simplifying expressions
When solving equations, you may need to simplify algebraic expressions as much as
possible.
For example:
3(5𝑥 + 1) simplifies to 3 ∙ 5𝑥 + 3 ∙ 1
and
10𝑥 − 3𝑥 simplifies to 7𝑥
We say that 3(5𝑥 + 1) is equivalent to 3 ∙ 5𝑥 + 3 ∙ 1, and 10𝑥 − 3𝑥 is equivalent to 7𝑥.
1)
Which expressions are equivalent to the expression below? There may be more
than one correct answer.
2(5𝑥 + 4) – 3𝑥 + 1
2)
a)
10𝑥 + 8 – 3𝑥 + 1
b)
18𝑥 – 3𝑥 + 1
c)
15𝑥 + 1
d)
7𝑥 + 9
Which expressions are equivalent to the expression below? There may be more
than one correct answer.
−3(2𝑥 – 1) + 3(𝑥 + 1) − 4𝑥
a)
−6𝑥 + 3 + 3𝑥 + 3 − 4𝑥
b)
−10𝑥 + 6 + 3𝑥
c)
−𝑥 + 12
d)
7𝑥 + 6
Exponents
It is important to pay careful attention to notation when working with negatives. For
example, – 52 is not the same as (– 5)2 . You can verify this by evaluating each
Copyright © 2020, The Charles A. Dana Center at The University of Texas at Austin
Foundations of Mathematical Reasoning Version 3.0 (2020)
Student Pages
expression on a calculator. The order of operations is the reason for this difference. The
negative in each expression can be thought of as multiplication by – 1. Look at the
expressions rewritten with a – 1 and think about what operation you would perform first.
−1 ∙ 52
(−1 ∙ 5)2
In the first expression, the exponent is done first: −1 ∙ 52 → −1 ∙ 25 → −25.
In the second expression, the multiplication is done first: (−1 ∙ 5)2→ (– 5)2 → 25.
3)
Simplify each of the following:
Part A: 52 =
Part B: −32 =
Part C: (−4)2 =
Part D: −42 =
Part E: −(−6)2 =
4)
Simplify the following expression:
−52 − (−3)2 + 62 =
𝑥
5)
Solve 5 = 4 − 3
6)
Solve
7)
Solve
8)
Solve 2(5𝑥– 3) = −42
𝑥+7
2
=6
𝑥−14
4
= 2𝑥
In algebra, a term is an individual part of an algebraic expression. Terms are separated
by + or – signs, and can consist of numbers, variables (letters), or the product of
numbers, and one or more variables. In instances where a number and variables are
Copyright © 2020, The Charles A. Dana Center at The University of Texas at Austin
Foundations of Mathematical Reasoning Version 3.0 (2020)
Student Pages
being multiplied, the number is called a coefficient. For example, the expression below
has three terms as shown in brackets.
−𝑎3 + 2𝑏 − 5
[−𝑎3 ] + [2𝑏] + [−5]
The first two terms are called variable terms because the value of the term changes
based on the value of the variable.
The coefficient of −𝑎3 is −1 because it can be written as (−1)( 𝑎3 ).
The coefficient of 2𝑏 is 2.
The last term, – 5, is called a constant term because its value never changes.
Notice that in the original expression the constant term was written as minus 5, but this
was rewritten as adding negative 5. When breaking an expression into terms, ask
yourself, “What is being added?”
9)
State the number of terms in each expression.
Part A: 3𝑥 + 4 has ______ term(s).
Part B: 5𝑥 − 4𝑥 2 + 2 has ______ term(s).
Part C: 5 has _____ term(s).
10) What is the coefficient of the 𝑥 2 term in 5𝑥 − 4𝑥 2 + 2?
11) What is the coefficient of the 𝑥 term in −3𝑥 2 − 2(−𝑥 + 4) + 5𝑥?
12) Recall the formula from In-Class Activity 12.D that was used to find Jenna’s cost to
drive her own car for work.
𝐽 = Cost of driving Jenna’s car in $/mile
𝑔 = Cost of gas in $/gallon
𝑔
𝐽=
+ 0.146
22
Part A: Find the cost of driving Jenna’s car (𝐽) when the price of gas is
$3.56/gallon. Round to the nearest hundredth.
Copyright © 2020, The Charles A. Dana Center at The University of Texas at Austin
Foundations of Mathematical Reasoning Version 3.0 (2020)
Student Pages
Part B: Find the cost of gas (𝑔) when the cost of driving Jenna’s car is $0.32/mile.
Round to the nearest hundredth.
Monitoring your readiness
13) To effectively plan and use your time wisely, it helps to think about what you know
and do not know. For each of the following, rate how confident you are that you
can successfully do that skill. Use the following descriptions to rate yourself:
5—I am very confident I can do this task.
4—I am somewhat confident I can do this task.
3—I am not sure how confident I am.
2—I am not very confident I can do this task.
1—I am definitely not confident I can do this task.
Before the next class, you should understand the concepts and demonstrate the
skills listed below.
Skills Needed for In-Class Activity 15.D
Skill or Concept: I can . . .
Apply the order of operations when
simplifying an expression.
Questions to check
your understanding
Rating
from 1 to 5
1–4, 12
Square a number.
3, 4
Solve a two-step linear equation such as 2 =
x/3 + 5.
5–8
Understand the meaning of the word term in
an algebraic expression.
9–11
Substitute a value for a variable in a
mathematical model and simplify the model.
12
Now use the ratings to get ready for your next in-class activity. If your rating is a 3 or
below, you should get help with the material before class. Remember, your
instructor is going to assume that you are confident with the material and will not take
class time to answer questions about it.
Copyright © 2020, The Charles A. Dana Center at The University of Texas at Austin
Foundations of Mathematical Reasoning Version 3.0 (2020)
Student Pages
Practice Assignment 15.B
1)
3
24
𝑥
6
Which of the following represent the same proportional relationship as 𝑥 = 56?
There may be more than one correct answer.
a)
b)
c)
d)
e)
2)
𝑥
24
3
= 56
3
𝑥
= 56
24
24
3
=
𝑥
56
𝑥
3
= 56
24
𝑥
3
=
56
24
Which of the following represent the same proportional relationship as 5 = 30?
There may be more than one correct answer.
a)
b)
c)
d)
e)
5
𝑥
6
𝑥
5
=
=
30
6
30
5
6
= 30
𝑥
𝑥
6
𝑥
3
5
= 30
3
= 15
𝑥
10
5
13
3
15
3)
Solve the proportion =
4)
Solve the proportion 4 = 2𝑥, rounding your answer to the nearest hundredth.
5)
Solve the proportion
6)
Cefaclor is a medication used for infections. It is often given in liquid form. A
pharmacist is mixing a prescription for a child. The instructions indicate that 125
mg of Cefaclor should be mixed with 5 ml of fluid. If the child only requires a
dosage of 100 mg of Cefaclor, how much fluid should be in each dosage?
24
7
=
, rounding your answer to the nearest hundredth.
120
𝑥
, rounding your answer to the nearest hundredth.
Copyright © 2020, The Charles A. Dana Center at The University of Texas at Austin
Foundations of Mathematical Reasoning Version 3.0 (2020)
Student Pages
7)
Ibuprofen is an over-the-counter medication often used for headaches and pains. It
is often given in pill form. A pharmacist typically recommends drinking 8 ounces of
water for every 400 mg of Ibuprofen taken orally. If a doctor recommends taking
1,000 mg of Ibuprofen to treat post-surgery pain, how many ounces of water
should a person drink?
8)
A company is making pennants or flags for a sports team. The team wants small
versions for fans and large versions that will fly over the stadium. The dimensions
of the small version are shown below.
8 in.
(base)
20 in.
(height)
Part A: Place the following items in the equation to make a true proportion.
small base
small height
large base
large height
=
Part B: The large version needs to be 12.5 feet across the base. How long should
it be?
Part C: How much material will be needed to make the large version of the flag?
Round to the nearest tenth of a square foot.
[Continued on the next page.]
Copyright © 2020, The Charles A. Dana Center at The University of Texas at Austin
Foundations of Mathematical Reasoning Version 3.0 (2020)
Student Pages
The following problem should be printed out, completed, and placed in your
binder.
9)
Now try another question about Cefaclor, a medication, often used in liquid form,
that is used for infections. A pharmacist is mixing a prescription for a child. The
instructions from the manufacturer indicate that 125 mg of Cefaclor should be
mixed with 5 ml of fluid. The pharmacist is mixing a prescription for 14 doses. For a
child this size, each dose should contain 80 mg of Cefaclor. How much fluid will be
in the full prescription?
Show your work thoroughly. Write a memo to the pharmacist with information
about the amount of fluid that should be included in the full prescription.
Copyright © 2020, The Charles A. Dana Center at The University of Texas at Austin
Foundations of Mathematical Reasoning Version 3.0 (2020)
Student Pages
Practice Assignment 15.E
𝑥
256
1)
Solve 4.5 =
2)
Solve 3𝑥 =
3)
Solve
4)
Lance Berkman of the Houston Astros had 180 hits in 520 at-bats. What is the ratio
of hits to at-bats? Be sure to simplify your answer.
5)
A standard HDTV screen with a width of 43.6 in has a height of 24.5 in. What is
the ratio of width to height of the HDTV? Be sure to simplify your answer.
6)
Chris spends 17 hours in a two-week period practicing the guitar. If he wants to
keep his practice time proportional to this ratio over the next five weeks, how many
hours should he practice?
7)
In a shipment of 800 tablets, 28 are found to be defective. How many defective
tablets should be expected in a shipment of 2,000?
8)
If a piece of wire 17 centimeters in length weights 102 ounces, what will 20
centimeters of the same wire weigh?
35
2𝑥−1
14
.
36
175
15
.
9
= 18.
Copyright © 2020, The Charles A. Dana Center at The University of Texas at Austin