Sets & Venn Diagrams
Let A= {perfect squares} and B= {nonpositive integers}.
Find A\capB
A\capB={0}
Simplify.
-2(2.14)-\abs{-2(2.14)
-8.56
Write & solve a proportion:
The tax on a $24 restaurant meal was $1.44. Find the tax on a restaurant meal costing $35.
\frac{1.44}{24}=\frac{x}{35}
50.4=24x
2.1=x
The tax on a $35 meal is $2.10.
Express 56.0768 as a mixed number in lowest terms.
56.0768
=56\frac{768}{10000
=56\frac{48}{625}
In a group of 100 people, everyone speaks English and/or French but no other languages. 72 people can speak English and 43 can speak French. How many people speak English only?
72+43-100=15
speak both languages
72-15=57
speak only English.
True or false? Be prepared to explain!
For any real number x,
\abs x=x
False! Example: any negative number.
Express the following ratio in lowest terms:
0.62 : \frac{1}{500}
0.62 : \frac{1}{500}
=310:1
Express in scientific notation.
(3.8\times 10^11)\times (6.2\times 10^-89)
(3.8\times 10^11)\times (6.2\times 10^-89)
=23.56\times 10^-78
=2.356\times10^-77
Order using \subset:
\bbbQ, \bbbN, \bbbR, \bbbZ
\bbbN\sub\bbbZ\sub\bbbQ\sub\bbbR
Solve.
\abs{-p}+\abs{p}=\abs{3.2-9.7}
2\abs{p}=\abs{-6.5}
2\abs{p}=6.5
\abs{p}=3.25
p\in{-3.25,3.25}
Solve.
42/(3-2x)=-28/(2x+1)
x=-4.5
Express 6.2\overline{15} as a mixed number in lowest terms.
6.2\overline{15}=6\frac{71}{330}
Is the following set empty? If so, explain. If not, find at least one element in the set.
\bbbQ^c\cap\bbbR
Not empty! Any irrational number, for example:
\pi, e,\sqrt 2,\cdots
Solve.
-\abs{x}=\abs{-x}
-\abs{x}=\abs{-x}
-\abs{x}=\abs{x}
x=0
Daisy had $66 in savings, while her older brother Oliver had $82. One day, Daisy & Oliver each donated the same amount of money to charity. The ratio of their savings became 3:7. How much money do they each have left?
Let x= amount donated. Then,
\frac{66-x}{82-x}=\frac{3}{7}
(66-x)(7)=(82-x)(3)
462-7x=246-3x
462=246+4x
216=4x
54=x
Daisy has 66-54= $12 left, while Oliver has 82-54= $28 left.
Does the fraction \frac 121{34375} have a terminating or repeating decimal representation? Be prepared to explain!
Terminating! We have
\frac{121}{34375}
=\frac{11}{3125}
=\frac{11}{5^5}
Fill in the blank with always, sometimes, or never. Be prepared to explain!
If A, B are non-empty sets, A\cupB is ____ a subset of A
sometimes! true exactly when
B\subsetA
Solve.
\abs{5x+8}=\abs{2x-2}
5x+8=2x-2
or
5x+8=-(2x-2)
x\in{-3\frac{1}{3}, -\frac{6}{7}}
Answer using a proportion:
The ratio of the sum of two numbers to the difference between them is 5:1. What is the ratio of the two numbers?
\frac{a+b){a-b}=\frac{5}{1}
5(a-b)=a+b
5a-5b=a+b
4a=6b
\frac{a}{b}=\frac{6}{4}
\frac{a}{b}=\frac{3}{2}
True or false? A proper fraction with a denominator of 765 will have a repeating decimal representation.
(Be prepared to explain.)
False!
\frac{153}{765}=\frac 1 5 = 0.2