Evaluating Expressions, and Solving Equations & Inequalities

To evaluate an algebraic expression, you have to substitute a number for each variable and perform the arithmetic operations.

Example

Evaluate the following expression for x = 7, y = 4 and z = 6

4z+(9−y) +6x

What you would need to do first is to replace the variable with the numbers given for the expression. Then solve the expression using order of operations.

4(6) + (9 - 4) + 6(7)

24 + 5 + 42 = 71

An equation is a statement that states that the values of two mathematical expressions are equal. This is represented by an equal sign between the two expressions.

When we have an equation with an unknown variable it is called an Algebraic Equation

To solve an Algebraic equation you must get the variable by itself on one side of the equation in order to solve for the unknown variable. This can be done by doing the following steps depending on the type of equation:

• Adding or subtracting the same value from both sides
• Clear out any fractions by Multiplying every term by the bottom parts
• Divide every term by the same nonzero value
• Combine Like Terms
• Factoring
• Expanding 
• Recognizing a pattern
• Apply a function to both sides 

Once you solve for the variable make sure you check your solution in the original problem to make sure it equals the same value on both sides of the equation.

Example: 

3x - 7 = 11

In this equation you will need to get the X by itself, to do this we would need to do the inverse operation of subtraction by adding seven to both sides of the equation.

3x - 7 + 7 = 11 + 7

This would give us the equation:     

 3x = 18

This is because when you add 7 to an - 7 they cancel each other out on the left side of the equation. then 11+ 7 = 18

Now to solve 3x = 18, you must divide each side by three because you want to do the inverse operation of the original equation which was multiplying by three. 

3x / 3 = 18 / 3

when we divide by three it would leave us with x = 6.

Now that we know x = 6, we must now put it back into the equation in order to check our solution. We would replace the x variable with the number six in the equation.

3(6) - 7 = 11

18 - 7 = 11

11 = 11

This means are solution was correct because both sides equal the same value

An inequality is an mathematical relationship between two expressions that is represented by using one of the following symbols:

≤: "less than or equal to"

≥: "Greater than or equal to"

<: "less than"

≠: "not equal to" 

>: "greater than"

To solve an inequality you would solve it the same way as an equation. Only exception is if you multiply or divide by a negative number the original symbol is turned towards the opposite direction. For instance, if the original symbol was like this > then it would be flipped to this <. Another way a symbol would change is if you swapped left and right hand sides. For instance, if you ended up with 7 > x and you flipped it around then you would put x < 7.

Example:

5 - 2x ≤ 7

First step is to get the x variable by itself in the inequality by subtracting five from each side of the inequality. 

 5 - 5 - 2x  ≤ 7 - 5

This would give us: - 2x ≤ 2. Now we must divide both sides by -2 in order to get X by itself. 

-2x / -2 ≤ 2 / -2.

Since we divided by a negative number are symbol would flip directions.

x ≥ -1

Two check our solution we would plug the -1 into the X position. 

5 - 2(-1) ≤ 7 

5 + 2 ≤ 7

7 ≤ 7