Mathematical Relationships and Basic Graphs - Algebra 2

To solve:

1. To clear the absolute value sign, we must split the equation to two possible solution. One solution for the possibility of a positive sign and another for the possibility of a negative sign:

or

2. solve each equation for :

or

Solve the following equation for b:

Let's begin by subtracting 6 from both sides:

Next, we can get rid of the absolute value signs and make our two equations:

Next, add 13 to both sides

And divide by 5:

or

Next, add 13 to both sides

And divide by 5:

So our answers are

Since the absolute value function only produces positive answers, an absolute value can never be equal to a negative number.

When dividing radicals, check the denominator to make sure it can be simplified or that there is a radical present that needs to be fixed. Since there is a radical present, we need to eliminate that radical. However, there is a faster way to possibly elminate the denominator. Let's simplify the numerator.

We still need to eliminate the radical so multiply top and bottom by .

.

To solve this you need to set up two different equations then solve for x.

The first one is:

where

The other equations is:

where

Recall the general definition of absolute value,

We will attempt two cases for the absolute value,

Case 1:

Start by isolating the abolute value term,

Replace with and solve for :

Case 2:

Replace with and solve for :

Testing the Solutions

Whenever solving equations with an absolute value it is crucuial to check if the solutions work in the original equation. It often occurs that one or both of the solutions will not satisfy the original equation.

For instance, if we test in the original equation we get,

This is clearly not true, since both sides are not equal, this rules out as a solution. Similiarly, using you can show that it also fails. Therefore, there is no solution.

If , then or

Solve each of those equations to find the possible values for x.

or

When dealing with absolute value equations, we need to deal with negative values as well.

Subtract on both sides.

Distribute the negative sign to each term in the parenthesEs.

Add on both sides.

Divide both sides by .

Answers are

When dealing with absolute value equations, we need to deal with negative values as well.

Add on both sides.

Subtract on both sides.

Distribute the negative sign to each term in the parentheses.

Add on both sides.

Divide on both sides.

Answers are

When dealing with absolute value equations, we need to deal with negative values as well.

Subtract on both sides.

Divide on both sides.

The coupon will entitle him to one free croissant, so Jerry will pay for the large espresso, the large cappucino, and one butter croissant. The charge will be the sum of the three prices:

Use the distance formula to solve this problem.

Plugging in our points A and B we get the following distance.

First, take the first equation and solve for x in terms of y.

From here, substitute the equation you found for x into the second equation and solve for y.

Now, substitute the y value found into the original equation and solve for x.

Simplify first by eliminating the parentheses. To do this you will need to distribute the coefficient term outside the parentheses to each term within the parentheses.

Next, combine like terms.

There is only one like-term in the expression given, which is .

Add the like term. The rest of the terms cannot be combined by adding or subtracting.

The correct answer is:

Add the ones digits.

Since this number is or greater, use the tens digit as a carry over.

Add the tens digits with the carry over.

Add the hundreds digit with the carry over. The hundreds digit for the second number in the problem is zero.

Since there is no carry over and there is no thousands digit for the second number, the thousands digit of the answer is just .

Combine all the ones digits in chronological order from thousands to the ones digits.

The answer is:

In order to simplify this, we need to combine like-terms. The coefficients of unlike terms cannot be combined.

Add and .

Add and .

Add and .

Combine all the terms.

The answer is:

In order to solve the problem, first evaluate .

Borrow a one from the six in the tens digit in order to subtract the ones digit.

The six in the tens digit becomes a five. Subtract the five with the tens digit of the second number.

Add .

Add the ones digit.

Since the number is ten or greater, use the tens digit as a carryover to the next calculation.

Combine the ones digits.

The answer is .