4|3x - 1| + 1 = 8
4|3x - 1| = 8 - 1
4|3x - 1| = 7
|3x - 1| = 7/4
Now lets think, we have 3x - 1 in module, so either its 7/4 or -7/4, we will have 7/4 at the end because of it, so we may have 2 solutions in this case:
3x - 1 = 7/4
and
3x - 1 = -7/4
So let's see:
3x - 1 = 7/4
3x = 7/4 + 1
3x = 11/4
x = 11/12
3x - 1 = -7/4
3x = -7/4 + 1
3x = -3/4
x = -1/4
So we have two possible answers, x = 11/2 and x = -1/4
The answer is 5265 milliliters
The answer is <span>mean = 13,027; median = 12,200; no mode
</span>
Let's rearrange values from the lowest to the highest:
11350, 12050, 12200, 13325, 16211
<span>The mean is the sum of all values divided by the number of values:
</span>(11350 + 12050 + 12200 + 13325 + 16211)/5 ≈ 13027
The median is the middle value. If there is an odd number of data, then the median is the value in the middle. In the data set 11350, 12050, 12200, 13325, 16211, the median (the middle value) is 12200
<span>The mode is the value that occurs most frequently. Since none of the number does not occur most frequently, there is no mode.
</span>
<span>binomial </span>is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.
We sometimes need to expand binomials as follows:
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
<span>(a + b)4</span> <span>= a4 + 4a3b</span><span> + 6a2b2 + 4ab3 + b4</span>
<span>(a + b)5</span> <span>= a5 + 5a4b</span> <span>+ 10a3b2</span><span> + 10a2b3 + 5ab4 + b5</span>
Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.
Pascal's Triangle
We note that the coefficients (the numbers in front of each term) follow a pattern. [This was noticed long before Pascal, by the Chinese.]
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
You can use this pattern to form the coefficients, rather than multiply everything out as we did above.
The Binomial Theorem
We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases.
<span>Properties of the Binomial Expansion <span>(a + b)n</span></span><span><span>There are <span>\displaystyle{n}+{1}<span>n+1</span></span> terms.</span><span>The first term is <span>an</span> and the final term is <span>bn</span>.</span></span><span>Progressing from the first term to the last, the exponent of a decreases by <span>\displaystyle{1}1</span> from term to term while the exponent of b increases by <span>\displaystyle{1}1</span>. In addition, the sum of the exponents of a and b in each term is n.</span><span>If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.</span>
