**Objectives**

- Isolate the variables in an equation or inequality by identifying the operations involved.
- Collect like terms to simplify the equation or inequality.
- Remember the order of operations for solving an equation or inequality in one variable in the proper sequence.
- Identify equations that have no solution or an infinite number of solutions.
- Ask the questions necessary to turn real-life problems into mathematical sentences and equations in one variable.

**Key Terms**

- Distributive Property – The rule that if a, b, and c are numbers or expressions, then a(b + c) = ab + ac.
- Example
- Step 1: Start with the original equation
- 3(x + 4) – 2x = 5

- Step 2: Distribute (using multiplication)
- 3x + 12 – 2x = 5

- Step 3: Collect like terms
- (3x – 2x) + 12 = 5

- Step 4: Simplify
- x + 12 = 5

- Step 5: Solve
- X = -7

- Step 1: Start with the original equation

- Example

- Infinite – Without end or limit; going on forever; impossible to count.
- Like Terms – Terms in an algebraic expression that have the same variables raised to the same powers.
- 5ab and -22ab are like terms because they have the same variables in each term.
- 8y and -5x are not like terms because they have different variables.

- Linear Equations – Equations whose graphs are straight lines. A linear equation can be written in the form y = mx + b.

**Notes**

- To solve these types of equations, you have to “undo” the variable expression. First you have to isolate the variable term, and then you have to isolate the variable itself.
- To undo (cancel) addition (positives), you must subtract terms on BOTH sides.
- To undo (cancel) subtraction (negatives), you must add the terms to BOTH sides.
- To undo (cancel) multiplication, you must divide terms or coefficients on BOTH sides.
- To undo (cancel) division, you must multiply terms or coefficients on BOTH sides.

- No Solution
- When an equation has no solution, there is no value for
*x*that would make the equation true. - Example: for
*x*=*x*+ 7, no matter what value you use for*x*, you will always get the untrue statement 0 = 7.- Start with the equation: x = x + 7
- Subtract x on both sides: 0 = 7
- 7 does not equal zero! So, this is untrue; therefore, no solution!

- When an equation has no solution, there is no value for

- Infinite Solutions
- If you try to solve an equation that has an infinite number of solutions, you will end up with an equation that is always true, no matter what the value of x is.
- Example: if x = x is the answer (or if two sides of the equation are identical), every solution is possible; so, infinite solutions is the answer.
- Start with the equation: 3 (x + 2) = 3x + 6
- Distribute 3 into the parenthesis: 3x + 6 = 3x + 6
- Notice how the two sides are identical!

- Subtract 6 from both sides: 3x = 3x
- Divide by 3 on both sides: x = x
- So, for ANY value you substitute into x, both sides will ALWAYS be equal.

- Example: 3x + 4 = 13
- Step 1: Subtract 4 from both sides to isolate the variable term
- 3x = 9

- Step 2: Divide both sides by 3 to isolate the variable
- x = 3

- Step 1: Subtract 4 from both sides to isolate the variable term

- Example: 3x – 4 = 17
- Step 1: Add 4 to both sides to isolate the variable term
- 3x = 21

- Step 2: Divide both sides by 3 to isolate the variable
- x = 7

- Step 1: Add 4 to both sides to isolate the variable term

- Example
- Step 1: Start with the equation
- Step 2: Distribute (multiply 1/2 into each term inside the parenthesis)
- Step 3: Collect like terms
- Step 4: Simplify (combine like terms)
- Step 5: Add 2x to both sides
- Step 6: Subtract 2 from both sides
- Step 7: Divide both sides by 4 and write the answer

- Step 1: Start with the equation