Solving Two-Step Equations



Solving Two-Step Equations 3-1

A two step equation contains two operations. To solve, use inverse operations to undo each operation in reverse order, opposite the order of operations. First, undo addition and or subtraction. Then, undo multiplication and or division.

Example 1 Example 2 Example 3

Example 4 Example 5 Example 6

The previous example could be done like this.

Solving Equations Having Like Terms and Parentheses 3-2

In order to solve equations with like terms, you must first simplify the equation by combining like terms, the same way you previously simplified expressions. After the equation has been simplified, you solve using inverse operations and begin working backwards, opposite the order of operations.

Example 1 Example 2 Example 3

If the question contains parentheses in addition to like terms that need to be simplified, first use distributive property to get rid of the parentheses. Then, follow the same process you used in examples one through three.

Example 4 Example 5 Example 6

Find the value of x. since

Example 7 then

Perimeter = 48 units

4

x + 7

Solving Equations with Variables on Both Sides 3-3

To solve equations with variables on each side, use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side of the equation. Then, solve the equation.

Example 1 Example 2 Example 3

Example 4

Twice a number is 60 more than five times the number. What is the number?

Example 5

Eight less than three times a number equals the number. What is the number?

3-3 continued

Equations often contain grouping symbols such as parenthesis or brackets. The first step in solving these equations is to use the Distributive Property to remove the grouping symbols.

Example 6 Example 7

[pic]

Example 8

Find the perimeter of the square.

Since this is a square, the sides are equivalent.

3x – 1

2x + 4

So, the length of one side would be and the perimeter would be

3-3 continued

Some equations have no solution. That is, there is no value for the variable that would result in a true sentence. For such an equation, the solution set is called the null or empty set, and is represented by the symbol [pic].

Example 9

Ten never equals nine. This is NEVER true.

So the solution is [pic].

Other equations may have every number as the solution. An equation that is true for every value of the variable is called an identity.

Example 10

Since -1 always equals -1, the solution is all numbers. In other words, it doesn’t matter what number you use for x, any number would result in a true statement.

Solving Inequalities Using Addition or Subtraction 3-4

A mathematical sentence that contains the symbols < (less than), > (greater than), [pic] (less than or equal to), or [pic] (greater than or equal to) between two expressions is called an inequality. For example, the statement that the speed limit is 45mph can be shown by the sentence s[pic]45, which implies you can go 45 miles per hour or slower. Inequalities with variables are called open sentences. The solution of an inequality with a variable is the set of all numbers that the variable could be replaced with that would result in a true statement.

* Hint: When writing a verbal description as an inequality, keep in mind the following; when you see these phrases, use these symbols

no more than or at most [pic]

no less than or at least [pic]

The solutions of inequalities can be graphed on a number line. For example,

|a > 4 or 4 < a |a < 4 or 4 > a |

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|open circle and darken to the right |open circle and darken to the left |

|a [pic]4 or 4 [pic] a |a [pic] 4 or 4 [pic]a |

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| | |

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| | |

|closed circle and darken to the right |closed circle and darken to the left |

Some inequalities can be solved by using the Addition and Subtraction Properties of Inequalities. These properties say that when you add or subtract the same number from each side of an inequality, the inequality remains true.

Example 1 Example 2 Example 3

3-4 continued

Example 4

Since the solutions to an inequality include all rational numbers that satisfy it, inequalities have an infinite number of solutions.

Remember that the solutions for an inequality can also be graphed.

Example 5

This is the graph of the solution from example 2 above;

[pic]

Example 6

This is the graph of the solution from example 4 above;

[pic]

Solving Inequalities Using Multiplication or Division 3 – 5

Some inequalities can be solved by using the Multiplication and Division Properties of Inequalities. These properties say that when multiplying or dividing each side of an inequality by the same positive number, the inequality remains true. In such cases, the inequality symbol does not change.

Example 1 Example 2 Example 3

[pic] [pic] [pic]

When multiplying or dividing each side of an inequality by a negative number, the inequality symbol must be reversed.

Example 4 Example 5 Example 6

[pic] [pic] [pic]

3 – 5 continued

Example 7 Example 8

[pic] [pic]

Write the verbal sentence as an inequality. Then solve the inequality.

Example 9

16 is greater than twice a number.

[pic]

Example 10

The quotient of a number and – 4 is no more than 32.

[pic]

Solving Multistep Inequalities 3 – 6

When solving inequalities that involve more than one operation, work backward to undo the operations, just as when you solve multi-step equations.

Remember to reverse the inequality symbol if you multiply or divide each side of an inequality by a negative number, as in example 2.

Example 1 Example 2

Sometimes you will need to use the Distributive Property to begin simplifying inequalities that contain grouping symbols before you solve.

Example 3

[pic]

3 – 6 continued

In some questions you will have to begin by eliminating a division step by multiplying by the divisor.

Example 4

[pic]

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