Lesson 701 After completing this lesson you will be able to say I can use substitution to determine whether a given number in a specified set makes an equation or inequality true Key Terms ID: 538879
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Slide1
Solutions to Equations and Inequalities
Lesson
7.01Slide2
After completing this lesson, you will be able to say:
I
can
use substitution to determine whether a given number in a specified set makes an equation or inequality true.Slide3
Key Terms
Equation:
mathematical sentence that shows two expressions
are
equal using the equal sign
Solution:
Any value substituted for a variable that makes the
mathematical
sentence
trueSlide4
Example of an equation and solutionSlide5
Balancing an equation
You can find solutions to an equation by using a balance scale.
When an equation is balanced the scales are equal on both sidesSlide6
Balancing the Scale
The balance scale is not balanced, what can you do to balance the scale?
If we remove one block from the left side, the scale will be balanced
An equation is two expressions that are equal to each other. Just as you balanced the scales, you were proving that the left side was equal to the right side. Therefore, you created a true statement. For example, 4 = 4 is a true statement, whereas, 5 =
4
is a false statement.Slide7
Balancing the scales
You can determine if an equation is true or false by substituting a value in for the variable.
When the left side equals the right side, the equation is balanced. This means the equation is true.
When the left side and right side are not equal, the equation is unbalanced. This means the equation is false.Slide8
Balancing the scales
3 + 4 = 7 is a true statement.
7 = 7
4 + 4 = 7 is a false statement.
8 ≠ 7
Therefore, 3 is the only solution that makes the statement true.Slide9
Try it
Is 4 a solution to the equation 5x = 20?Slide10
Check your work
Check by substituting 4 for the variable and simplifying.
5(4) = 20 Substitute the variable with the given value and simplify.
20 = 20 Is this a true statement?
Yes! Therefore, 4 is a solution of the equation 5x = 20, because it makes a true statement.Slide11
Inequalities
Inequality:
A mathematical sentence that shows a comparison between two expressions using the less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) symbols.Slide12
Inequalities
The inequality symbols “is less than or equal to” (≤) and “is greater than or equal to” (≥) are like two symbols in one.
Think of a statement that uses one of these symbols as a combination of an inequality and an equation. If either the inequality or the equation is true, then the entire statement is true.Slide13
Inequalities - Examples
Is x ≤ 8 true, when x equals 5?
The statement 5 ≤ 8 is true if
either
the statement 5 = 8 or the statement 5 < 8 is true.
5 = 8 is false.
5 < 8 is true.Because 5 < 8 is true, the inequality 5 ≤ 8 is true because 5 is less than 8.
The solution can be less than or equal to as it cannot be both.Slide14
Inequalities - ExampleSlide15
Try It!
Barnabas believes that x = 7 is a solution to the inequality 4x + 5 > 34. Is he correct?Slide16
Check your work
4x + 5
>
34
4(7) + 5 > 34 Substitute 7 into the variable
of
the inequality.28 + 5 > 34 Simplify.33 >
34 Is this a true statement?This is not a true statement because 33 is not greater than 34. Therefore, 7 is not a solution of the inequality.Slide17
Try It
Why is x = 4 not a valid solution to the inequality
7x
+ 5 > 33?Slide18
Check your work
When you substitute x = 4 into the inequality and simplify, the statement is not true.
7x + 5
7(4) + 5 > 33 Substitute and simplify.
33
>
33 Is this a true statement?Because 33
is not greater than 33, x = 4 is not a solution of the inequality.Slide19
Sets of Numbers
Because inequalities
compare
two expressions, there are multiple values that can make the statement true. Sometimes, you may have to check multiple values that are presented in a set.Slide20
Sets of Numbers - Example
Which value or values from the set {1, 3, 5} make the inequality 4x + 8 > 12 a true statement? How do you know?
Substitute each value from the set into the inequality to see which values make a true statement.
Substitute
1 into the inequality and simplify
4x + 8 > 12
4(1) + 8 > 12
4 + 8 > 1212 > 12Is this a true statement?This is
not a true statement. The value 12 is not greater than 12, so x = 1 is not a solution.Substitute 3 into the inequality and simplify4x + 8 > 124(3) + 8 > 1212 + 8 > 1220 > 12Is this a true statement?
This is a true statement. The value 20 is greater than 12,
so x = 3 is a solution.
Substitute
5 into the inequality and simplify
4x + 8 > 12
4(5) + 8 > 12
20 + 8 > 12
28 > 12
Is this a true statement?
This is a true statement. The value 28 is greater than 12,
so x = 5 is a solution.Slide21
Try itSlide22
Check your work
Substitute
25 into the inequality and simplify
Is this a true statement?
This is
not
a true statement.
so x = 25 is not a solution.Substitute 45 into the inequality and simplify
Is this a true statement?This is a true statement. so x = 45 is a solution.Substitute 55 into the inequality and simplify
Is this a true statement?
This is a true statement. so x = 55 is a solution.Slide23
Try it!
Erica is creating a rectangular garden with an area less than or equal to 100 square feet. Erica can use the inequality LW ≤ 100 represent the area of the garden, where L is the length and W is the width. If the length of the garden has to be 25 feet, can she make the garden 5 feet wide?Slide24
Check your work
Substitute L = 25 and W = 5 into the inequality.
25(5) ≤ 100 Substitute and simplify.
125 ≤ 100 Is this a true statement?
This is not a true statement. Because 125 is not equal to or less than 100, Erica cannot make the garden 5 feet wide.Slide25
Now that you completed this lesson, you should be able to say:
I
can
use substitution to determine whether a given number in a specified set makes an equation or inequality true.