How to write expressions and equations - Algebra

When we see is less than, we need to use an inequality sign. The sign is . Our expression is .

It's much easier to use factoring and canceling than it is to use long division for this problem. 9_x_2 – 1 is a difference of squares. The difference of squares formula is a_2 – b_2 = (a + b)(a – b). So 9_x_2 – 1 = (3_x + 1)(3_x – 1). Putting the numerator and denominator together, (9_x_2 – 1) / (3_x_ – 1) = (3_x_ + 1)(3_x_ – 1) / (3_x_ – 1) = 3_x_ + 1.

Take every word and translate into math. Sum is addition. Is means equal sign. So we have .

Take every word and translate into math. Is less than means we need to use the symbol. We should have .

For a vertical line e.g. , and

This line does not intersect the and hence there is no .

Since the line passes through hence the -intercept .

These two equations represent the same line and they "cross' at infinitely many points. Therefore these systems are dependent.

The function is not defined when and .

It is defined between

and

Take every word and translate into math.

Product is multiplication.

So we have as the answer.

Take every word and translate into math. Is greater than means we need to use the symbol. We should have .

Take every word and translate into math. Is less than or equal to means we need to use the symbol. We should have .

Let's start with the first half of the sentence, "eight less than three times a number". We can translate the written English into mathematical terms in the order that we read them. "Less than" means that we are going to be subtracting something (in this case ) from another value. That other value is "three times a number", which means multiplied by a number. We aren't told what this number is, but that doesn't matter, because we can just use instead. Now let's put together that first half of the sentence in math terms:

The next part of the sentence, "is equal to", tells us that we need to insert an equal sign into the equation next:

Now let's look at the last part of the sentence, "four more than two times that same number". "More than" means that we are going to be adding something (in this case ) to another value. That other value is "two times that same number", which lets us know that multiplcation is now involved. We are going to use that same number we used in the first half the sentence, we chose , and we're going to multiply that number by . Now let's put together the complete second half of the sentence in math terms:

And we can now place this on the other side of the equal sign to get a complete mathematical expression:

Break up the sentence into parts.

Twice a number:

The sum of twice a number and fifty:

The answer is:

We can simplify the natural log exponents by using the following rules for naturla log.

Using these rules, we can perform the following steps.

Knowing that the e cancels the exponential natural log, we can cancel the first e.

Distribute the square into the parentheses and calculate.

Remember that a negative exponent is equivalent to a quotient. Write it as a quotient and then you're finished.

A complex number in its standard form is of the form: , where stands for the real part and stands for the imaginary part. The symbol stands for .

The real part in this problem is 1.

Multiply both the numerator and the denominator by the conjugate of the denominator which is resulting in

This is equal to

Since you can make that substitution of in place of in both numerator and denominator, leaving:

When you then cancel the negatives in both numerator and denominator (remember that , simplifying each term), you're left with a denominator of and a numerator of , which equals .

Rewrite, then distribute:

Three less than five times a number is the same as two more than twice that number.

Let the number be .

"Three less than five times a number" translates into .

"Is the same as" means equal to or "".

"Two more than twice that number" means .

Putting these together gives:

Here is the expression given: .

To simplify, follow the order of operations.

Distribute through the terms in the inner parentheses:

Now distribute into the terms of the remaining parentheses. Remember that multiplied by itself produces , but multiplied by produces :

Complete the multiplication to finish expanding:

Add like terms to reach the answer:

Writing this expression in simplest terms can be achieved by first factoring the radical into its smallest factors.

Multiplying the two together results in . Multiplying this by (which is simplified to ) results in the answer .