Solve the inequality |4x +4| < 12

Mathematics

2 answers:

mel-nik [20]4 years ago

5 0

Start by dividing all terms by 4, to reduce this expression: |x+1| < 3.

Case 1: x+1 is already positive. The abs. val. operator does not change

anything. Then x+1<3, or x<2.

Case 2: (x+1) is negative. The abs. val. operator, in effect, puts a - sign in

front of (x+1): -(x+1)<3. Distributing the - sign: -x - 1 < 3. Adding 1 to both sides: -x < 4. We want to solve for positive x, so we mult. both sides of

-x<4 by -1 AND change the direction of the inequality symbol:

x > -4

Thus, the solution set is (-4,2): All real numbers between -4 and 2.

lana [24]4 years ago

5 0

$|4x +4| < 12\\\\ 4|x+1|$

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Need help on the answer

lions [1.4K]

Answer:

The second choice is the answer.

Step-by-step explanation:

In math, there are two types of numbers: real numbers and imaginary numbers. Imaginary numbers build upon the premise that the equation

x^2 +1 = 0 has no real solution. It is said that one of the imaginary solutions to this equation is <em>i</em>, or √-1. Real numbers are numbers that can represent the "distance" of a line. Real numbers have subsets known as natural numbers, whole numbers, integers, natural numbers. It also has a region of irrational numbers.

Natural numbers are numbers you use to count w/ your fingers (1, 2, 3 ...)

Whole numbers are natural numbers and 0 (0, 1, 2, 3 ...)

Integers are whole numbers and their opposites ( -1, 0, 1, ...)

Rational numbers integers and numbers that can be put into a fraction of two integers. for example, 1/2 is a fraction of the integers 1 and 2 ( -1, -1/2, 0, 1/2, 1 ...)

Irrational numbers are numbers that cannot be written as a simple fraction or cannot be written as terminating or repeating decimals (√2, π, <em>e, ...</em>).

The answer is not the first choice because it has whole numbers.

The answer is not the third choice because it has integers.

The answer is not the fourth choice because it has irrational numbers.

The answer is not the fifth choice because it has imaginary numbers.

The second choice is the one that has rational numbers and is the answer.

4 0

3 years ago

(HMMT Combo 2009) Two jokers are added to a 52 card deck and the entire stack of 54 cards is shuffled randomly. What is the expe

myrzilka [38]

The expected number of cards from the card deck that will be strictly between the two jokers is $\frac{52}{3}$ .

The most popular deck of cards used today is a normal 52-card deck of French-suited playing cards. [a] It is the only standard pack[b] used for playing cards in English-speaking nations; but, in many other nations throughout the world, it is used alongside other traditional, frequently older, standard packs with various suit systems, like those with German-, Italian-, Spanish-, or Swiss suits. The English pattern pack is the only one widely accessible in Britain and America and the most popular pattern of French-suited cards globally. The Belgian-Genoese pattern, which was created in France but is also used widely in Spain, Italy, the Ottoman Empire, the Balkans, and much of North Africa and the Middle East, is the second most popular.

Let all pairs of the positions (m,n) of the jokers are equally likely

$P[0]=P[|m-n|=1]=\frac{53}{\left(\begin{array}{c}54 \\2\end{array}\right)}=\frac{2}{27}$

$P[1]=P[|m-n|=2]=\frac{52}{\left(\begin{array}{c}54 \\2\end{array}\right)}=\frac{104}{1431}$

$P[k]=P|m-n|=k+1=\frac{54-k-1}{\left(\begin{array}{c}54 \\2\end{array}\right)}$

the expected value will be

$\begin{array}{l}\sum_{k=1}^{52} \frac{54-k-1}{\left(\begin{array}{c}54 \\2\end{array}\right)} k \\=\frac{1}{1431}\left(\sum_{k=1}^{52} 53 k-k^{2}\right) \\=\frac{1}{1431}\left(\frac{53}{2}(52)(53)-\frac{1}{6}(52)(53)(105)\right)\\=\frac{52}{3}\end{array}$