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`