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Section 2: Equations and Inequalities
Topic 1: Equations: True or False? ..................................................................................................................................... 29
Topic 2: Identifying Properties When Solving Equations ................................................................................................ 31
Topic 3: Solving Equations ................................................................................................................................................. 34
Topic 4: Solving Equations Using the Zero Product Property ......................................................................................... 36
Topic 5: Solving Inequalities - Part 1 ................................................................................................................................. 38
Topic 6: Solving Inequalities - Part 2 ................................................................................................................................. 40
Topic 7: Solving Compound Inequalities ......................................................................................................................... 43
Topic 8: Rearranging Formulas ......................................................................................................................................... 47
Topic 9: Solution Sets to Equations with Two Variables .................................................................................................. 49
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The following Mathematics Florida Standards will be covered in this section:
A-CED.1.1 - Create equations and inequalities in one variable and use them to solve problems. Include equations arising
from linear and quadratic functions, and simple rational and exponential functions.
A-CED.1. 2 - Create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales.
A-CED.1.4 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For
example, rearrange Ohm’s law! = #$ to highlight resistance $.
A-REI.1.1 - Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous
step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution
method.
A-REI.2. 3 - Solve linear equations and inequalities in one variable, including equations with coefficients represented by
letters.
A-REI.2.4 - Solve quadratic equations in one variable.
A-REI.4.10 - Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate
plane, often forming a curve (which could be a line).
A-SSE.1.2 - Use the structure of an expression to identify ways to rewrite it.
Let’s Practice!
- Consider the algebraic equation 6 + 3 = 9.
a. What value can we substitute for 6 to make it a true number sentence?
b. How many values could we substitute for 6 and have a true number sentence?
- Consider the algebraic equation 6 + 6 = 6 + 9. What values could we substitute for 6 to make it a true number sentence?
Try It!
- Complete the following sentences.
a. 8 ,^ = 4 is true for _________________________.
b. 29 = 9 + 9 is true for _________________________.
c. 8 + 67 = 8 + 68 is true for _________________________.
A number sentence is an example of an algebraic equation.
Ø An algebraic equation is a statement of equality between two __________________.
Ø Algebraic equations can be number sentences (when both expressions contain only numbers), but often they contain ________________ whose values have not been determined.
Consider the algebraic equation 4 6 + 2 = 46 + 8.
Are the expressions on each side of the equal sign equivalent? Justify your answer.
What does this tell you about the numbers we can substitute for 6?
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Section 2 – Topic 2
Identifying Properties When Solving Equations
The following equations are equivalent. Describe the operation that occurred in the second equation.
3 + 5 = 8 and 3 + 5 − 5 = 8 − 5
6 − 3 = 7 and 6 − 3 + 3 = 7 + 3
2 (4) = 8 and
,(>) ,
=
. ,
; ,
= 3 and 2 ⋅
; ,
This brings us to some more properties that we can use to write equivalent equations.
BEAT THE TEST!
- Which of the following equations have the correct solution? Select all that apply.
; < ,
= 6 + 5; 6 = 18
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Try It!
- The following pairs of equations are equivalent. Determine the property that was used to write the second equation.
a. 2 6 + 4 = 14 − 66 and 26 + 8 = 14 − 66
b. 26 + 8 = 14 − 66 and 26 + 8 + 66 = 14 − 66 + 66
c. 26 + 8 + 66 = 14 and 26 + 66 + 8 = 14
d. 86 + 8 = 14 and 86 + 8 − 8 = 14 − 8
e. 86 = 6 and +. ⋅ 86 = +. ⋅ 6
BEAT THE TEST!
- For each algebraic equation, select the property or properties that could be used to solve it.
Algebraic Equation Addition or Subtraction
Property of Equality Multiplication or Division
Property of EqualityDistributive Property Commutative Property ; ,
= (^5) o o o o
26 + 7 = 13 (^) o o o o
46 = 23 (^) o o o o
6 − 3 = − (^4) o o o o
4(6 + 5) = 40 (^) o o o o
10 + 6 = 79 (^) o o o o
−8 − 6 = 19 (^) o o o o
2(6 − 8) + 76 = 9 (^) o o o o
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Other times, a word problem or situation may require you to write and solve an equation.
A class is raising funds to go ice skating at the Rink at Campus Martius in Detroit. The class plans to rent one bus. It costs $150.00 to rent a school bus for the day, plus $11.00 per student for admission to the rink, including skates.
What is the variable in this situation?
Write an expression to represent the amount of money the school needs to raise.
The class raised $500.00 for the trip. Write an equation to represent the number of students who can attend the trip.
Solve the equation to determine the number of students who can attend the trip.
Section 2 – Topic 3
Solving Equations
Sometimes you will be required to justify the steps to solve an equation. The following equation is solved for 6. Use the properties to justify the reason for each step in the chart below.
Statements Reasons
a. 5 6 + 3 − 2 = 2 − 6 + 9 a. Given
b. 56 + 15 − 2 = 2 − 6 + 9 b.
c. 56 + 15 − 2 = 2 + 9 − 6 c.
d. 56 + 13 = 11 − 6 d. Equivalent Equation
e. 56 + 13 − 13 = 11 − 13 − 6 e.
f. 56 = −2 − 6 f. Equivalent Equation
g. 56 + 6 = −2 − 6 + 6 g.
h. 66 = −2 h. Equivalent Equation
i. F F; = E F, i.
j. 6 = −
= j.^ Equivalent Equation
BEAT THE TEST!
- The following equation is solved for 6. Use the properties to justify the reason for each step in the chart below.
Statements Reasons
a. 2 6 + 5 − 3 = 15 a. Given
b. 26 + 10 − 3 = 15 b.
c. 26 + 7 = 15 c. Equivalent Equation
d. 26 + 7 − 7 = 15 − 7 d.
e. 26 = 8 e. Equivalent Equation
f. ,;, = ., f.
g. 6 = 4 g. Equivalent Equation
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Section 2 – Topic 4
Solving Equations Using the Zero Product Property
If someone told you that the product of two numbers is 10 , what could you say about the two numbers?
If someone told you that the product of two numbers is zero, what could you say about the two numbers?
This is the zero product property.
Ø If ?@ = 0, then either? = 0 or @ = 0.
Describe how to use the zero product property to solve the equation 6 − 3 6 + 9 = 0. Then, identify the solutions.
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Let’s Practice!
- Identify the solution(s) to 26 6 + 4 6 + 5 = 0.
- Identify the solution(s) to 26 − 5 6 + 11 = 0.
Try It!
- Michael was given the equation 6 + 7 6 − 11 = 0 and asked to find the zeros. His solution set was {−11, 7}. Explain whether you agree or disagree with Michael.
- Identify the solution(s) to 2 K − 3 ⋅ 6(−K − 3) = 0.
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Addition and Subtraction Property of Inequality
Ø If? > @, then? + A > @ + A and? − A > @ − A for any real number A.
Consider 26 − 1 + 2 > 6 + 1. Use the addition or subtraction property of inequality to solve for 6.
Let’s Practice!
- Consider the inequality 4 + 6 − 5 ≥ 10. Use the addition or subtraction property of inequality to solve for 6. Express the solution in set notation and graphically on a number line.
Consider the following graph.
Ø When the endpoint is a(n) ____________ dot or circle, the number represented by the endpoint _______ a part of the solution set.
Write an inequality that represents the graph above.
Write the solution set that represents the graph above.
Why is “or equal to” included in the solution set?
Just like there are properties of equality, there are also properties of inequality.
If 6 > 5, is 6 + 1 > 5 + 1? Substitute values for 6 to justify your answer.
Try It!
- Consider the inequality 46 + 8 < 1 + (26 − 5). Use the addition or subtraction property of inequality to solve for 6. Express the solution in set notation and graphically on a number line.
- Peter deposited $27 into his savings account, bringing the total to over $234. Write and solve an inequality to represent the amount of money in Peter’s account before the $27 deposit.
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Section 2 – Topic 6
Solving Inequalities – Part 2
Consider 6 > 5 and 2 ∙ 6 > 2 ∙ 5. Identify a solution to the first inequality. Show that this solution also makes the second inequality true.
Consider 6 > 5 and −2 ∙ 6 > −2 ∙ 5. Identify a solution to the first inequality. Show that this solution makes the second inequality false.
How can we change the second inequality so that the solution makes it true?
Consider −R > 5. Use the addition and/or subtraction property of inequality to solve.
BEAT THE TEST!
- Ulysses is spending his vacation in South Carolina. He rents a car and is offered two different payment options. He can either pay $25.00 each day plus $0.15 per mile (option A) or pay $10.00 each day plus $0.40 per mile (option B). Ulysses rents the car for one day.
Part A: Write an inequality representing the number of miles where option A will be the cheaper plan.
Part B: How many miles will Ulysses have to drive for option A to be the cheaper option?
Try It!
- Find the solution set to the inequality. Express the solution in set notation and graphically on a number line.
a. −6 (6 − 5) > 42
b. 4(6 + 3) ≥ 2(26 − 2)
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Section 2 – Topic 7
Solving Compound Inequalities
Consider the following options.
Option A: You get to play NBA 2K after you clean your room and do the dishes.
Option B: You get to play NBA 2K after you clean your room or do the dishes.
What is the difference between Option A and B?
Circle the statements that are true.
2 + 9 = 11 and 10 < 5 + 6
4 + 5 ≠ 9 and 2 + 3 > 0
0 > 4 − 6 or 3 + 2 = 6
15 − 20 > 0 or 2.5 + 3.5 = 7
- Stephanie has just been given a new job in the sales department of Frontier Electric Authority. She has two salary options. She can either receive a fixed salary of $500.00 per week or a salary of $200.00 per week plus a 5% commission on her weekly sales. The variable Y represents Stephanie’s weekly sales. Which solution set represents the dollar amount of sales that she must generate in a week in order for the option with commission to be the better choice?
A Y Y > $300. B Y Y > $700. C Y Y > $3,000. D Y Y > $6,000.
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- Write a compound inequality for the following graphs.
a. Compound inequality:
b. Compound inequality:
Be on the lookout for negative coefficients. When solving inequalities, you will need to reverse the inequality symbol when you multiply or divide by a negative value.
Try It!
- Graph the solution set to each compound inequality on a number line.
a. 6 < 1 or 6 > 8
b. 6 ≥ 6 or 6 < 4
c. −6 ≤ 6 ≤ 4
BEAT THE TEST!
- Use the terms and symbols in the table to write a compound inequality for each of the following graphs. You may only use each term once, but you do not have to use all of them.
76 < 2 or ≤ 36 + >
Compound Inequality:
Compound Inequality:
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- Write a compound inequality for the following graphs.
a. Compound inequality:
b. Compound inequality:
