Algebra - GRE Subject Test: Math

Since the absolute value with x in it is alone on one side of the inequality, you set the expression inside the absolute value equal to both the positive and negative value of the other side, 11 and -11 in this case. For the negative value -11, you must also flip the inequality from less than to a greater than. You should have two inequalities looking like this.

and

Add 5 to both sides in each inequality.

and

Divide by -4 to both sides of the inequality. Remember, dividing by a negative will flip both inequality symbols and you should have this.

and

The correct answer is and

The correct answer is and

Before expanding the quantity within absolute value brackets, it is best to simplify the "actual values" in the problem. Thus becomes:

From there, note that the absolute value means that one of two things is true: or . You can therefore solve for each possibility to get all possible solutions. Beginning with the first:

means that:

For the second:

means that:

Note that the two solutions can be connected by putting the inequality signs in the same order:

Because Absolute Value must be a non-negative number, there is no solution to this Absolute Value inequality.

The following absolute value inequality can be used to assess the bowling balls that are tolerable:

The correct answer is and

At this point, you've isolated the absolute value and can solve this problems for both cases, and . Beginning with the first case:

Then for the second case:

The correct answer is and

The Absolute Value Inequality that can assess which cell phones are tolerable is:

Step 1: Separate the equation into two equations:

First Equation:
Second Equation:

Step 2: Solve the first equation

Step 3: Solve the second equation

The solution is

Step 1: Evaluate .

Step 2. Evaluate

From the previous step, we already know what is.

is just multiplying by another

Step 3: Evaluate .

The expansion of is

Solution:

We can look at Pascal's Triangle, which is a quick way to do Binomial Expansion. We read each row (across, left to right)

For the first row, we only have a constant.
For the second row, we get .
...
For the 7th row, we will start with an term and end with a constant.

Step 1: We need to locate the 7th row of the triangle and write the numbers in that row out.

The 7th row is .

Step 2: If we translate the 7th row into an equation, we get:

. This is the solution.

Method One:

We will start expanding slowly, and we will end up at exponent

Step 1: Expand:

Step 2: Multiply by the product of . By doing this, we are now expanding .

Step 3: Multiply by again

.

After Step 4:
After Step 5:

After Step 6, the final answer is:

.

Method Two:

You can find the expansion of this binomial by using the Pascal's Triangle (shown below)

If you look at Row of the triangle above, the row that starts with .

We need to negate every nd term, as the answer in Method One has every even term negative.

We will still get the answer: .

Step 1: Expand

Step 2: FOIL the first two parentheses:

Step 3: Multiply the expansion in step 2 by :

The expanded form of is .

Step 1: Multiply

Step 2: Multiply the result in step 1 by

Step 3: Multiply the result of step 2 by

Simplify:

Expand by distributing each of the factors

Simplify

Simplify

Step 1: Let's start small, expand .

Step 2: Expand

Take the final answer of Step 1 and multiply it by ...

Step 3: Multiply again by to the final answer of Step 2...

Let's start with a smaller expansion:

We multiply the expansion of by :

Multiply again by :

Multiply by :