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1-3 Open Sentences
In this section we are going to define a mathematical
sentence and the algebraic term solution.
We will also solve equations and
inequalities using replacement sets and order of operations.
Algebra 1 by Gregory HaucaGlencoe adapted from presentations by Linda Stamper
Sentences have verbs and phrases do not.
PhrasesPhrases are translated into expressions. ex. 2x + 3
SentencesSentences are translated into equations and inequalities.
In section 1-1 we discussed the difference between phrases and sentences in English.
In algebra
An equation is a mathematical sentence with the equal sign between two expressions.
ex. 5 + 3 = 8,
An inequality has an inequality sign between them.ex. 3 - 2 > 0,
3x + 1 = 4
2x + 1 < 4x - 3
An equation or inequality is “open” if it contains a variable expression.ex. 7x – 2 =
8, 3(x + 2) < 8
Equations and inequalities can be either TRUE or FALSE
Examples: 2(5) + 3 = 13 10 + 3 =
13True
7 - 2 > 8 + 2
5 > 10False
13 = 13
5 > 8 + 2
Open sentences are neither true nor false until the variable(s) have been replaced by specific values and the open sentence simplified.
Example: x + 3 = 5
5 = 5True
(2) + 3 = 5
The value or values of the variable that make an equation or inequality true is called the solutionsolution.
The process of finding a value for the variable that results in a true statement is called solving.
This replacement value is called a solution. An equation or inequality may have one, many, or no solutions.
To solve an equation given a replacement set, substitute each value into the equation and simplify to determine if it makes the equation true.
Find the solution set for 6n + 7 = 37 for the replacement set {4, 5, 6, 7}
solution: n = 5
6n 7 37
6n 7 37
3743
37736
37766
377n6
3749
37742
37776
377n6
6 4 7 37
24 7 37 31 37
The value or values of the variable that make an equation true is the solutionsolution.
Solving Equations and Inequalities using Replacement Sets
6 5 7 37 30 7 37
37 37
Ex.1
The value or values of the variable that make an equation true is the solutionsolution.
Find the solution set for 5(x + 2) = 40 if the replacement set is {4, 5, 6, 7}
solution: n = 6
5 x 2 40
5 x 2 40
5 5 2 40
5 7 40
35 40
5 x 2 40
5 6 2 40
5 8 40
40 40
5 x 2 40
5 7 2 40
5 9 40
45 40
5 4 2 40
5 6 4030 40
Inequalities.
< is less than 4 < 8
< is less than or equal to 4 < 4 > is greater than 20 > 5
> is greater than or equal to 20 > 20
inequality symbol meaning example
What’s the importance of is in each of the above?
It is a verb. Without it you will have a subtraction or addition not an inequality.
n is less than 4
Learn to distinguish inequality from subtraction.
n less than 4
Expression (no verb)
Is this an expression or a
sentence?
4−n
Is this an expression or a
sentence?
Sentence (verb is)n < 4
To solve an inequality given a replacement set, substitute each value into the inequality and simplify to determine if it makes the inequality a true statement.
The value or values of the variable that make an inequality true is the solutionsolution.
Find the solution set for 30 + n > 37 if the replacement set is {5, 6, 7, 8}
solution: n = {7, 8}
30 n 37
3736
37630
37n30
3737
37730
37n30
3738
37830
37n30
30 5 37 35 37
The value or values of the variable that make an inequality true is the solutionsolution.
Ex.2 Find the solution set for 9 > 2y − 5 if the replacement set is {5, 6, 7, 8}
solution: y = {5, 6}
9 2y 5
79
5129
5629
5y29
99
5149
5729
5y29
119
5169
5829
5y29
9 2 5 5 9 10 5 9 5
Write the problem.Follow rules for order of operations.
Solving equations by applying the order of operations.
453
4213m
13 8m
21m
3
m 7
The value or values of the variable that make an equation true is the solutionsolution.
Solve.
3 1
Whiteboard Practice
129
y2
81y
3
y 27
1243
573x
2
9 2
x3 4 3
7
x12 3
7x
15
k3518
2853
3
5 10k
18 2
5 10k
18 8
50
k10
5 k
424
3425x 3
2
5 4 4 3x
4 8 4
20 4 3x
4 4
20 12x
16
32x
16
x 2
Find the solution set for 5x + 10 = 40 if the replacement set is {4, 5, 6, 7}
Solution: x = 6
5x 10 40
5x 10 40
5 5 10 40
25 10 40
35 40
5x 10 40
5 6 10 40
30 10 40
40 40
5x 10 40
5 7 10 40
35 10 40
45 40
5 4 10 40
20 10 40
30 40
HW1-A6 Pages 18-20 #14-30 even,34,36,53,57,64-67. Page 9 #49-54.
Remember! Correct your odd-numbered
homework problems in the back of the
book!