Download Lesson 11

Name: ____________________________________
1.3.2 Properties of Equalities and Potential Dangers
Date __________
Algebra 1
Essential Questions: How are the properties of equalities and inequalities used to find solution
sets and are there any "dangers" to be aware of?
Property of
Equality
Addition Property
of Equality
Subtraction
Property of
Equality
Multiplication
Property of
Equality
Division Property
of Equality
Algebraically
If a = b, then a
+ c = b + c 
If a = b, then a
-c=b-c
If a = b, then
a•c=b•c
If a = b, then
a b
=
c c
Description
Example
adding the same term to both
sides of an equation will not
change the equality
subtracting the same term from
both sides of an equation will not
change the equality
multiplying both sides of an
equation by the same term will
not change the equality
dividing both sides of an equation
by the same term will not change
the equality
Describe the property used to convert the equation, one line to the next.
3x – 7 = 14
3x = 21
x=7
Mathematical Process
_________________
_________________
Mathematical Property
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How do you know that the first line and the last line of the equation above holds true for
the same solution?
What is the solution? Is it true for all equations above?
1
4
Mathematical Process
Mathematical Property
𝑥 = 11
____________________
______________________
𝑥 = 44
____________________
______________________
𝑥 + 5 = 16
1
4
Addition/Subtraction Properties of Inequalities
𝐼𝑓 𝐴 > 𝐵 𝑡ℎ𝑒𝑛 𝐴 + 𝑐 > 𝐵 + 𝑐 𝑜𝑟 𝑖𝑓 𝐴 < 𝐵, 𝑡ℎ𝑒𝑛 𝐴 + 𝑐 > 𝐵 + 𝑐
𝐼𝑓 𝐴 > 𝐵 𝑡ℎ𝑒𝑛 𝐴 − 𝑐 > 𝐵 − 𝑐 𝑜𝑟 𝑖𝑓 𝐴 < 𝐵, 𝑡ℎ𝑒𝑛 𝐴 − 𝑐 > 𝐵 − 𝑐
The addition/subtraction property of inequality says adding/subtracting the same term
to both sides of an inequality will not change the solution set of the inequality.
Multiplication/Division Properties of Inequalities
𝐼𝑓 𝐴 > 𝐵 𝑡ℎ𝑒𝑛 𝑘𝐴 > 𝑘𝐵 𝑜𝑟 𝑖𝑓 𝐴 < 𝐵, 𝑡ℎ𝑒𝑛 𝑘𝐴 < 𝑘𝐵
𝐼𝑓 𝐴 > 𝐵 𝑡ℎ𝑒𝑛(−𝑘)𝐴 < (−𝑘)𝐵 𝑜𝑟 𝑖𝑓 𝐴 < 𝐵, 𝑡ℎ𝑒𝑛 (−𝑘)𝐴 > (−𝑘)𝐵
*Remember, when multiplying or dividing by a negative number, the order of the
inequality changes (flip the inequality symbol).
The multiplication/division property of inequality says multiplying/dividing by a term to
both sides of an inequality will not change the solution set of the inequality.
Describe the property used to convert the equation, one line to the next.
3𝑥 − 7 < 14
3𝑥 < 21
𝑥<7
Mathematical Process
____________________
____________________
Mathematical Property
______________________
______________________
1
Mathematical Process
Mathematical Property
________________________
__________________________
________________________
__________________________
4
𝑥 + 5 ≥ 16
1
4
𝑥 ≥ 11
𝑥 ≥ 44
Potential Dangers when solving equations
Using the equation 𝑥 − 4 = 3
1. Multiply both sides of the equation by a constant to show that the solution set doesn't change.
2. Now multiply both sides of the original equation by 𝑥 .
a) Show that 𝑥 = 7 is still a solution to the new equation.
b) Show that 𝑥 = 0 is also a solution to the new equation.
3. Using the new equation 𝑥(𝑥 − 4) = 3𝑥
Now multiply the new equation by the factor 𝑥 − 1
a) Show that x = 7 is still a solution to the equation.
b) Show that x = 1 is also a solution to the equation.
*Based on your results, what effect does multiplying an equation by one of the following have on
the solution set?
a) a constant?
b) a variable factor?
Closing:
1) What moves did we see that did not have any effect on the
solution set of an equation?
2) What moves did we see that did have an effect on the
solution set of an equation?