# College Algebra : Solving Equations and Inequallities

## Example Questions

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### Example Question #1 : How To Find A Solution Set

Give all real solutions of the following equation:

Explanation:

By substituting  - and, subsequently,  this can be rewritten as a quadratic equation, and solved as such:

We are looking to factor the quadratic expression as , replacing the two question marks with integers with product  and sum 5; these integers are .

Substitute back:

The first factor cannot be factored further. The second factor, however, can itself be factored as the difference of squares:

Set each factor to zero and solve:

Since no real number squared is equal to a negative number, no real solution presents itself here.

The solution set is .

### Example Question #1 : Solving Equations And Inequallities

Give all real solutions of the following equation:

The equation has no real solutions.

Explanation:

By substituting  - and, subsequently,  this can be rewritten as a quadratic equation, and solved as such:

We are looking to factor the quadratic expression as , replacing the two question marks with integers with product 36 and sum ; these integers are .

Substitute back:

These factors can themselves be factored as the difference of squares:

Set each factor to zero and solve:

The solution set is .

### Example Question #141 : Expressions & Equations

Solve for :

Explanation:

can be simplified to become

Then, you can further simplify by adding 5 and  to both sides to get .

Then, you can divide both sides by 5 to get .

### Example Question #1 : Solving Equations And Inequallities

Solve for :

Explanation:

To solve for , you must first combine the 's on the right side of the equation. This will give you .

Then, subtract  and from both sides of the equation to get .

Finally, divide both sides by  to get the solution .

### Example Question #1 : Linear Equations

Solve the following equation for :

Explanation:

The first step is to distribute (multiply) the 2 through the parentheses:

Then isolate  on the left side of the equation. Subtract the 10 from the left and right side.

Finally, to isolate , divide the left side by 2 so that the 2 cancels out. Then divide by 2 on the right side as well.

You can verify this answer by plugging the  into the original equation.

### Example Question #1 : Linear Equations

Solve for :

Explanation:

Combine like terms on the left side of the equation:

Use the distributive property to simplify the right side of the equation:

Next, move the 's to one side and the integers to the other side:

### Example Question #2 : Solving Equations And Inequallities

Solve for x:

Explanation:

Simplify the parenthesis:

Combine the terms with x's:

Combine constants:

### Example Question #2 : Linear Equations

Solve the following equation when y is equal to four.

Explanation:

Solve the following equation when y is equal to four.

To solve this equation, we need to plug in 4 for y and solve.

### Example Question #7 : Solving Equations And Inequallities

Solve the following:

Explanation:

To solve, we must isolate x. In order to do that, we must first add 7 to both sides.

Next, we must divide both sides by 3.

### Example Question #1 : Linear Equations

Write an equation of the line passing through (5,10) and (10,2).