Equations / Inequalities - Algebra

Through factoring, the sum of the two roots must equal 2, and the product of the two roots must equal . , and 3 satisfy both of these criteria.

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form.

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First, create two blank binomials.

Start by factoring our first term back into the first term of each biniomial. Since the only reasonable roots of are and , we know that

Next, factor out our constant, ignoring the sign for now. The factors of are either and , and , or and , We must select those factors which have either a difference or a sum equal to the value of in our trinomial. In this case, neither and nor and can sum to , but and can. Now, we can add in our missing values:

One last step remains. We must check our signs. Since is positive in our trinomial, Either both signs are negative or both are positive. To figure out which, check the sign of in our trinomial. Since is negative, both signs in our binomials must be negative.

Thus, our two binomial factors are and . Note that this can also be written as , and if you graph this, you get a result identical to .

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form, but this is already done. Note that this is a special trinomial: , and thus only and are present to be manipulated.

First, create two blank binomials.

Start by factoring our first term back into the first term of each biniomial. Since the only reasonable roots of are and , we know that

Next, factor out our constant, ignoring the sign for now. The only factors of are either and , or and , We must select those factors which have either a difference or a sum equal to the value of in our trinomial. In this case, only and sum to , which we use since is not present.

One last step remains. We must check our signs. Since is negative in our trinomial, one sign is positive and one is negative.

Thus, our two binomial factors are and .

This is called a difference of squares. If you see a trinomial in the form , the roots are always and .

Subtract 2 from each side:

This inequality can be solved just like an equation.

Add 4 to both sides:

2x > 11

Then divide by 2:

x > 11/2 = 5.5

First, combine like terms on a single side of the inequality. On the right side of the inequality, combine the terms to obtain .

Next, we want to get all the variables on the left side of the inequality and all of the constants on the right side of the inequality. Add 4 to both sides and subtract from both sides to get .

Finally, to isolate the variable, divide both sides by 12 to produce the final answer, .

First, add to both sides:

, or

.

Then, add 2 to both sides:

, or

Finally, divide both sides by 6 to get the answer:

which simplifies to:

To solve the inequality, simply move the 's to one side and the integers to the other (i.e. subtract from both sides and add 9 to both sides). This gives you .

Subtracting and adding 3 to both sides of the equation of will give you . Divide both sides by 2 to get .

Add and subtract 2 from both sides of to get . Then, divide both sides by 3 to get a solution of . The only answer choice that is greater than 5 is 6.

Note the switch in inequality symbols when the numbers are multiplied by a negative number.

or, in interval notation,

Combine like-terms on the left side of the inequality: . Next, isolate the variable: .

Therefore the answer is .

To solve , isolate the variable by adding three on both sides.

The correct answer is:

To solve the inequality, get all terms with on one side and all constants on the other side. We first subtract from both sides

,

Now add 7 to both sides

.

Now divide both sides by 2

To solve this inequality, we need to separate the constants from the variables so that they are on opposite sides of the inequality.

We can do this by adding (4x+5) to each side and

.

The constants cancel on the left side, and the variables cancel on the right side.

Then, we divide both sides by 16, to get our final answer:

This is a one-step problem in which all you need to do is add the to both sides to get by itself.

So,

Then simplify to get:

To solve an inequality, isolate the variable on one side with all other constants on the other side. To accomplish this, perform opposite operations to manipulate the inequality.

First, isolate the x by subtracting three from each side.

Whatever operation you do to one side you must do to the other side as well.

This gives you:

The answer, therefore, is .

In order to isolate the variable, we will need to add nine on both sides.

Simplify both sides of the equation.

The answer is .

Add both sides by four to isolate the variable.

Simplify both sides of the equation.

The answer is:

This means that is greater than forty, but cannot equal to forty.

When solving an inequality, we will solve it the same way we would solve an equation. We are solving for x, so we want x to stand alone. In the equation

we want to add 5 to both sides. The inequality symbol does not change. We get