Solving Equations and Inequalities

Chapter 7

Solving Two-Step Equations

Section 7-1

How to Solve a Two-Step Equation

• One goal in solving an equation is to have only variables on one side of the equal sign and numbers on the other side of the equal sign.

• The other goal is to have the number in front of the variable equal to one. – The variable does not always have to be

x. These equations can make use of any letter as a variable.

How to Solve Two-Step Equations continued…

The most important thing to remember in

solving a linear equation is that whateveryou do to one side of the equation, youMUST do to the other side.

– So if you subtract a number from one side, you MUST subtract the same value from the other side. You will see how this works in the examples.

Solving Two-Step Equations continued…

Solving a two-step equation requires thesame procedure(s) as a one-step

equation. However, the order in which the

proceduresare done makes a difference.

   

Do the inverse operation for addition or subtraction first.

Do the inverse operation of multiplication or division last.

Inverse The

operation that

reverses the effect of another operation.

Solving Multi-Step Equations

Section 7-2

Solving the Equation

Solve 3x – 9 = 33 Step 1: You need to get the first term with the variable by itself. So you need to “UNDO” or get rid of -9. To do this, you do the OPPOSITE of what is being done. You are currently SUBTRACTING 9; the OPPOSITE is ADDITION. What you do to one side, you MUST do to the other. So you will ADD 9 to both sides.

Step 2: The variable is still not by itself, so you need to do the OPPOSITE of what is being done. 3 is being multiplied to X and the OPPOSITE of multiply is DIVIDE. So you DIVIDE both sides by 3.

The X is finally alone which means you have completed the equation.

Finding Consecutive Integers

• The definition of consecutive is “following one another in uninterrupted intervals.”

• When you count by 1’s from any integer, you are counting consecutive integers.

1, 2, 3, 4 -5, -4, -3

The sum of three consecutive integers is 96Let n = the least integer.Then n + 1 = the second integer,and n + 2 = the third integer.

n + n + 1 + n + 2 = 96

Four consecutive integers Three consecutive integers

Steps for Solving a Multi-Step Equation

Step 1: Use the Distributive Property, if necessary.

Step 2: Combine like terms.Step 3: Undo addition or subtraction.Step 4: Undo multiplication or

division.

Let’s Practice

Multi-Step Equations with Fractions and

DecimalsSection 7-3

Solving Multi-Step Equations with Fractions

To clear a fraction from an

equation, you multiplyboth sides of the

equationby the denominator.

Examples 4x=12 2n-6=22 5

3

-7k+14=-21 1x+3=210

4

How to Clear Equations of Fractions

TO SOLVE AN EQUATION WITH fractions, wetransform we transform it into an equation

withoutfractions – which we know how to solve. Thetechnique is called clearing of fractions.

Example 1   

Solve for x:x + x − 2 = 63       5 

Multiply both sides of the equation--every term--by the LCM of denominators.  Every denominator will then cancel.  We will then have an equation without fractions. The LCM of 3 and 5 is 15.  Therefore, multiply every term on both sides of the equal sign by 15. Each denominator will now cancel into 15--that is the point--and we have the following simple equation that has been "cleared" of fractions.

Caution! Be sure the distributive law is used to

multiply all of the terms by 15.

How to Clear Equations of Decimals

To clear an equation of decimals, we count the

greatest number of decimal places in any onenumber. If the greatest number of decimal

placesis 1, we multiply both sides by 10; if it is 2, wemultiply by 100; and so on.

Example 2

Solve: 16.3 - 7.2y = -8.18

Solution The greatest number of decimal places in any one number is two. Multiplying by 100 will clear alldecimals.

Quick Check

Solve each equation:

a. -7 + y = 1 b. 1b – 1 = 5 12 6 3 6

c. 1.5x – 3.6 = 2.4 d. 1.06p – 3 = 0.71

Write an Equation

Section 7-4

Steps to Writing Equations

• Read the problem carefully and figure out what it is asking you to find. – Usually, but not always, you can find this information at the end of the

problem. • Assign a variable to the quantity you are trying to find.

– Most people choose to use x, but feel free to use any variable you like. • Write down what the variable represents.

– By the time you read the problem several more times and solve the equation, it is easy to forget where you started.

• Re-read the problem and write an equation for the quantities given in the problem. – The only way to truly master this step is through lots of practice. Be

prepared to do a lot of problems. • Solve the equation. • Answer the question in the problem.

– Just because you found an answer to your equation does not necessarily mean you are finished with the problem. Many times you will need to take the answer you get from the equation and use it in some other way to answer the question originally given in the problem.

• Check your solution. – Your answer should not only make sense logically, but it should also

make the equation true.

Solving Equations with Variables on Both Sides

Section 7-5

How to Solve an Equation with Variables on Both

Sides• Consider the equation x – 6 = –2x + 3. To

isolate the variable, we need to get all the variable terms to one side and the constant terms to the other side. – You may need to use the Distributive Property to

simplify one or both sides of an equation before you can get the variable alone on one side.

• The first step is to use addition or subtraction to collect the variable on one side of the equation.

• Next, we combine like terms and then isolate the variable by multiplying or dividing.

Let’s Practice…Example:

Solve x – 6 = –2x + 3

Solution:

Step 1: Get all the variable terms to one side and the constant terms tothe other side.

x – 6 = –2x + 3

Step 2: Combine like terms

2x + x = 3 + 6

3x = 9

Step 3: Divide or multiply to isolate

the variable

3x = 9

Check:

x – 6 = –2x + 3