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Multistep Equations

How to Identify Multistep Equations |Combining Terms| How to Solve Multistep Equations | Consecutive Integers | Multistep InequalitiesMultistep Equation How to Identify Multistep Equations | Combining Terms | How to Solve Multistep Equations | Consecutive Integers | Multistep Inequalities1Learning ObjectivesUse the properties of equality to solve multistep equations of one unknownApply the process of solving multistep equations to solve multistep inequalities

Multistep EquationsAfter this lesson, you will be able to use the properties of equality to solve multistep equations of one unknown, and apply the process of solving multistep equations to solve multistep inequalities.

2Some equations can be solved in one or two stepsEx) 4x + 2 = 10 is a two-step equationSubtract 2Divide both sides by 4Ex) 2x 5 = 15 is a two-step equationAdd 5Divide both sides by 2

How to Identify Multistep Equations

Some equations can be solved in one step, and some require two.

The equation 4x + 2 = 10 {four x plus two equals ten} is a two-step equation.

Solving it requires first subtracting 2 from both sides and then dividing both sides by 4. Although this equation requires two steps to solve, there is only one unknown term (x).

The equation 2x 5 = 15 {two x minus five equals fifteen} is another example of a two-step equation.

The variable x appears only once, and solving for this variable requires only two steps.

This image shows a sliding block puzzle. There are many ways that the blocks can be scrambled, and solving this type of puzzle requires many steps. However, the process of solving sliding block puzzles can be learned quickly because they are always solved the same way. {play animation, read following sentence as animation begins}. Following a certain sequence of steps leads to the solution regardless of how the puzzle was scrambled.

3Multistep equation an equation whose solution requires more than two steps Ex) 5x 4 = 3x + 2 and 4(x 2) = 12Multistep equations can take different formsVariables present in two different termsEx) 6x 2x = 8 + 4, 5x 4 = 3x + 2Ex) 4(x 2) = 12

How to Identify Multistep Equations

CharacteristicExample Multiple variable or constant terms on the same sidex + 2x + 3x 1 = 3 + 20Variable present on both sides4x 2 = 3x + 3Parentheses present on either side3(x + 2) = 21Like the sliding block puzzle, some equations require many steps in order to solve. A multistep equation is an equation whose solution requires more than two steps.

One example is the equation 5x 4 = 3x + 2 {five x minus four equals three x plus two}. Notice that although there is only one unknown (x), it is present on both sides of the equation. These two occurrences of the variable can be combined, but doing this requires extra steps.

The equation 4(x 2) = 12 {four times x minus two equals twelve} is another example of a multistep equation. In this example, the distributive property must be used in order to simplify the left side of the equation. As a result, this equation requires more than two steps to solve.

Although multistep equations can take different forms, there are certain features that indicate when an equation requires multiple steps in order to solve.

One is if the variable is present in two different terms.

These terms could be on the same side of the equation, as in 6x 2x = 8 + 4 {six x minus two x equals eight plus four}, but frequently they are on opposite sides, as in 5x 4 = 3x + 2 {five x minus four equals two x plus two}.

Another feature that can indicate a multistep equation is the presence of parentheses on either side of it, as in 4(x 2) = 12 {four times x minus two equals twelve.

Some characteristics that indicate that an equation may require more than two steps are listed.

4First step to solving a multistep equation is to simplify each sideUse the distributive property to eliminate parenthesesEx) 5(x 2) + x = 2(x + 3) + 4 simplifies to 5x 10 + x = 2x + 6 + 4 Combine like termsEx) 5x 10 + x = 2x + 6 + 4 Combine 5x and x to 6x on the left sideCombine 6 and 4 to 10 on the right sideSimplifies to 6x 10 = 2x + 10Terms containing variables cannot be combined with constant terms

Combining Terms

The first step in solving a multistep equation is to simplify each side.

This means using the distributive property, if necessary, to eliminate parentheses.

Both sides of the equation 5(x 2) + x = 2(x + 3) + 4 {five times x minus two plus x equals two times x plus three plus four} contain parentheses that can be eliminated in this way. The distributive property simplifies this equation to 5x 10 + x = 2x + 6 + 4 {five x minus ten plus x equals two x plus six plus four}.Simplifying each side also involves combining like terms.

On the left, the terms 5x and x can be combined to 6x since each is a multiple of the same variable.

On the right, the constant terms 6 and 4 can be combined.

This produces the simplified equation 6x 10 = 2x + 10 {six x minus ten equals two x plus ten}.

It is important to remember that terms containing the variable cannot be combined with constant terms.

5Consider 5(x 3) + 4 = 3(x + 1) 2 Distributive Property can be used on both sides due to the parentheses multiplied by constantsProduces 5x 15 + 4 = 3x + 3 2 Terms can be combined on both sidesCombine 15 and 4 on the left and 3 and 2 on the rightProduces 5x 11 = 3x + 1Combining TermsAnother example of a problem that requires combining like terms is the equation 5(x 3) + 4 = 3(x + 1) 2 {five times x minus three plus four equals three x plus one minus two}.

The numbers multiplied by expressions in parentheses make it clear that the distributive property must be used on both sides of this equation.

Using the distributive property on each side produces 5x 15 + 4 = 3x + 3 2 {five x minus fifteen plus four equals three x plus three minus two}.

This new equation has terms on each side that can be combined, the 15 and 4 on the left and the 3 and 2 on the right.

This simplifies the equation to 5x 11 = 3x + 1 {five x minus eleven equals three x plus one}.

6Consider the equation 4(x + 2) 10 = 2(x + 4) Simplify with the distributive property4x + 8 10 = 2x + 8Combine like terms4x 2 = 2x + 8Eliminate the unknown from one side2x 2 = 8Eliminate the constant term on the other side2x = 10Divide each side by the coefficient of the variablex = 5How to Solve Multistep EquationsMultistep equations can usually be solved by a straightforward process. The first step is to simplify each side by eliminating grouping symbols and combining like terms. Consider the equation 4(x + 2) 10 = 2(x + 4) {four times x plus two minus ten equals two times x plus four}. The presence of parentheses, as well as the apparent complexity of the equation, indicates that this is a multistep equation.

Each side can be simplified by the distributive property, producing the form 4x + 8 10 = 2x + 8 {four x plus eight minus ten equals two x plus eight}. The left side can now be simplified further because it contains two constant terms.

Combining these simplifies the equation to 4x 2 = 2x + 8 {four x minus two equals two x plus eight}.The equation is now simpler in form than the original, but the unknown (x) still appears on both sides. It is easier to solve for the unknown if it occurs only once in the equation.

As a result, the second step of solving a multistep equation is to eliminate the unknown from one side of the equation. It can be eliminated from either side, but since the variable is traditionally written on the left, it is more common to eliminate it from the right side of the equation.

In this example, subtracting 2x from each side of the equation eliminates the variable from the right side, leaving 2x 2 = 8. This is now a two-step equation since the variable occurs only once, and only two operations are performed on it, namely multiplication and subtraction. As a result, it can now be solved as a two-step equation by undoing each of these operations.

The next step is to eliminate the constant term on the side with the variable.

Adding 2 to both sides yields 2x = 10.

The final step is to divide both sides by the coefficient of the variable.

Dividing each side of this equation by 2 results in the solution, x = 5.

7Same general set of steps is useful in solving many multistep equations

How to Solve Multistep Equations

Steps to Solve a Multistep EquationFirst stepSimplify each side as much as possibleSecond stepEliminate the variable from one sideThird stepEliminate the constant term from the side with the variableFourth stepDivide each side by the coefficient of the variable

Solving this equation required only four basic steps.

Although multistep equations can appear in many different forms, the same general set of steps is useful in solving many of them.

After identifying the problem as a multistep equation, apply the following steps. First, simplify each side of the equation as much as possible by combining like terms and using the distributive property if necessary in order to remove parentheses. Second, eliminate the variable on one side of the equation. Third, eliminate the constant term on the side with the variable. Fourth, divide each side by the coefficient of the variable. Some multistep equations may not require every step. For example, if the variable has no coefficient after the second step is completed, then the fourth step is unnecessary.

8How to Solve Multistep Equations Example

Ex) Mark and Susan are given the same amount of money. Mark spends $5, and Susan spends $20. If Mark now has twice as much money as Susan, how many dollars did they each have originally? JustifyThey each began with $35Evaluate35 5 is double 35 20AnalyzeFind the original amountsFormulateRepresent the problem as an equationDetermine x 5 = 2(x 20) x 5 = 2x 40 x 5 = 40 x = 35 x = 35Mark and Susan are given the same amount of money. Mark spends $5, and Susan spends $20. If Mark now has twice as much money as Susan, how many dollars did they each have originally?

First, analyze the problem. The problem presents a word problem about Mark and Susan, who begin with equal amounts of money. When Mark spends $5 and Susan spends $20, Mark has twice as much as Susan. The problem asks for the original amount that they each had.

Next, formulate a plan or strategy to solve the problem. Represent the problem as an equation. Simplify each side, eliminate the variable from one side and the constant from the other side, and divide both sides by the coefficient of the variable.

Next, determine the solution to the problem. Represent the problem as an equation. Use the distributive property. Subtract 2x from each side. Add 5 to each side. Divide each side by 1.

Now that the solution has been determined, justify it. An equation was written showing Marks amount (on the left) equal to twice Susans amount (on the right). The right side of the equation was simplified, and then the variable term was eliminated from the right, and the constant term was eliminated from the left. Dividing each side by the coefficient of x gave the solution, which shows that Mark and Susan each began with $35.

Last, evaluate the effectiveness of the steps, and the reasonableness of the solution. The steps for solving a multistep equation worked effectively. The answer is reasonable because 35 5 is double 35 20. 9Consecutive integers integers that are separated by exactly one unitEx) 5 and 6 Set of consecutive integers from 1 to 3When domain is limited to integers, the letter n is used to represent the variableEx) Find 3 integers that sum to 24

Solution is 7, 8, 9Consecutive Integers

Consecutive integers are integers that are separated by exactly one unit.

The numbers 5 and 6 are consecutive integers because there is no other integer in between them.

Here is a set of consecutive integers. Note that each number in this set is equal to the previous number plus 1.

When the domain of a variable is limited to integer values, the letter n is frequently used to represent that variable. Problems involving consecutive integers usually can be expressed as multistep equations. In these equations, the integer with the lowest value is usually written as n because this letter is generally used for variables that are restricted to integer values. Since the first integer is n, the second must be n + 1, the third being n + 2, and so forth.

For example, suppose that a problem asks for three consecutive integers whose sum is 24.

Since the three integers n, n +1, and n + 2 are added together, this problem can be written as the equation n + n + 1 + n + 2 = 24 {n plus n plus one plus n plus two equals twenty four}. This equation can be solved by following the sequence of steps for solving multistep equations.

The three integers in this situation would be 7, 8, and 9.

10Ex) Find three consecutive integers where the sum of the first two is equal to 3 times the third oneCan be represented as n + (n + 1) = 3(n + 2)Can be solved using the process to solve multistep equationsConsecutive IntegersAs another example, suppose a problem asks for three consecutive integers where the sum of the first two is equal to 3 times the third one.

This problem can be represented by the equation n + (n + 1) = 3(n + 2) {n plus n plus one equals three times n plus two}.

Again, this equation can be solved by following the sequence of steps for solving multistep equations.

11Consecutive Integers Example

Ex) Mrs. Smith has three children whose ages are spaced one year apart. How old will they be when their ages add to 45?

Justify

Evaluate14, 15, and 16 are consecutive integers and add to 45Analyze

FormulateRepresent the problem as an equationDeterminen + (n + 1) + (n + 2) = 45n + n + 1 + n + 2 = 453n + 3 = 453n = 42n = 14, n + 1 = 15, n + 2 = 16

Mrs. Smith has three children whose ages are spaced one year apart. How old will they be when their ages add to 45?

First, analyze the problem. The problem presents three children whose ages are consecutive integers. It asks for all of their ages when their sum is 45.

Next, formulate a plan or strategy to solve the problem. Represent the problem as an equation, and solve it using the steps for solving multistep equations. Use the solution to identify the age of each child.

Next, determine the solution to the problem. Represent the problem as an equation. Remove parentheses according to the associative property. Combine like terms. Subtract 3 from each side. Divide each side by 3. Identify the ages of the other children.

Now that the solution has been determined, justify it. The three consecutive integers were represented in an equation as n, n +1 , and n +2. Since the solution of the equation was n = 14, the three integers were 14, 15, and 16.

Last, evaluate the effectiveness of the steps, and the reasonableness of the solution. Representing the problem as an equation provided a straightforward way to solve it using the steps for solving multistep equations. The answer is reasonable because the numbers 14, 15, and 16 are consecutive integers and add to 45. 12Consecutive even integers are spaced two units apartEx) 6, 8, and 10Can be represented as n, n + 2, n + 4Consecutive odd integers represented the same wayEx) 7, 9, and 11Can be represented as n, n + 2, n + 4Check answer to make sure the solutions are odd

Consecutive Integers

Some problems ask for consecutive even integers or consecutive odd integers. Consecutive even integers are not one unit apart because any even number plus 1 is an odd number. As a result, consecutive even integers are spaced two units apart.

The numbers 6, 8, and 10 are consecutive even integers.

Any set of three consecutive even integers can be represented as n, n + 2, and n + 4.

Consecutive odd integers, such as 7, 9, and 11, can also be represented as n, n + 2, and n + 4 because they are also spaced two units apart.

Even and odd integers are represented the same way in an equation.

As a result, if a problem asks for a set of consecutive odd integers, it is important to check the answer at the end to make sure that the solutions obtained are actually odd.

13Consecutive Integers Example

Ex) Four brothers sit next to each other at a baseball game, and they notice that the seat numbers are all odd in their section. If their seat numbers add to 80, what is the greatest of the seat numbers?

Justify

Evaluate Reasonable as the sum of the four integers 17, 19, 21, and 23 adds to 80AnalyzeFind the greatest seat numberFormulateRepresent the problem as an equation and solve itDeterminen + (n + 2) + (n + 4) + (n + 6) = 80n + n + 2 + n + 4 + n + 6 = 804n + 12 = 804n = 68n = 17, so n + 6 = 23

Four brothers sit next to each other at a baseball game, and they notice that the seat numbers are all odd in their section. If their seat numbers add to 80, what is the greatest of the seat numbers?

First, analyze the problem. The problem describes four people whose seat numbers form a set of consecutive odd integers whose sum is 80. It asks for the greatest of these four numbers.

Next, formulate a plan or strategy to solve the problem. Represent the problem as an equation, and solve it using the steps for solving multistep equations. Use the solution to identify the greatest of the four numbers.

Next, determine the solution to the problem. Represent the problem as an equation. Use the associative property to remove parentheses. Combine like terms. Subtract 12 from both sides. Divide both sides by 4. Add 6 to both sides.

Now that the solution has been determined, justify it. The four consecutive odd integers were represented in an equation as n, n + 2, n + 4, and n + 6. Since the first integer was found to be n = 17, the greatest one (n + 6) was found to be 23.

Last, evaluate the effectiveness of the steps, and the reasonableness of the solution. Representing the problem as an equation provided a straightforward way to solve it using the steps for solving multistep equations. The answer is reasonable because it is the largest odd integer from the sequence 17, 19, 21, 23. The sum of the four odd integers also adds to 80. 14Multistep inequalities can be solved in the same way as multistep equationsEx) 6(x 2) > 2x can be solved with the four step process for multistep equationsFirst, simplify each side, 6x 12 > 2xSecond, eliminate variable on right side, 4x 12 > 0 Third, eliminate constant from left side, 4x > 12 Last, divide each side by coefficient of unknown, x > 3Ex) 2(x + 6) 6xFirst, simplify each side, 2x + 12 6xSecond, eliminate variable on right side, 4x + 12 0Third, eliminate constant from left side, 4x 12Last, divide each side by coefficient of unknown, x 3

Multistep InequalitiesTwo-step inequalities can be solved in the same way as two-step equations. Similarly, multistep inequalities can be solved in the same way as multi-step equations.

For example, the inequality 6(x 2) > 2x {six times x minus two is greater than two x} can be solved with the four-step process for solving multistep equations.

First, the distributive property simplifies the left side, producing 6x 12 > 2x {six x minus twelve is greater than two x}.

Second, subtracting 2x from each side eliminates the variable from the right side, resulting in the inequality 4x 12 > 0 {four x minus twelve is greater than zero}.

Third, adding 12 to both sides eliminates the constant from the left side, giving 4x > 12 {four x is greater than twelve}.

Finally, dividing each side by 4 produces the solution, x > 3 {x is greater than three}.

Recall that dividing both sides of an inequality by a negative number causes the inequality sign to flip. Consider, for example, the multistep inequality 2(x + 6) {two times x plus six} less than or equal to 6x.

The distributive property simplifies the inequality to 2x + 12 6x {two x plus twelve is less than or equal to six x}.

Subtracting 6x from each side produces 4x + 12 0 {negative four x plus twelve is less than or equal to zero}, and subtracting 12 from each side produces 4x 12 {negative four x is less than or equal to twelve}.

When each side is divided by 4, the inequality sign flips, giving the result x 3 {x is greater than or equal to three}.

15Sometimes it is simpler to put the variable term on the right side instead of the leftEx) 2x + 12 6x can be made into a two-step inequalitySubtract 2x from each side to produce 12 4xDivide by 4 to yield 3 x Placing the variable on the left also requires that the inequality sign be flipped, resulting in x 3Numbers in the solution range, such as 5, can illustrate that the statements 3 x and x 3 are equivalentMultistep Inequalities

It may be simpler in some problems to put the variable terms on the right side instead of the left.

For example, 2x + 12 6x {two x plus twelve is less than or equal to six x} can more quickly be made into a two-step inequality by subtracting 2x from each side, yielding 12 4x {twelve is less than or equal to four x}, and dividing by 4, yielding the inequality 3 x {three is less than or equal to four}.

Switching the sides in order to place the variable on the left also requires that the inequality sign be flipped, resulting in x 3 {x is greater than or equal to three}.

A number in the solution range, such as 5, can illustrate that the statements 3 x {three is less than or equal to x} and x 3 {negative x is less than or equal to negative three} are both equivalent to x 3 {x is greater than or equal to three}.

16Multistep Inequalities Example

Ex) Find the solution set of the inequality 3(x + 2) > 5x. Justify

Evaluatex < 3 describes a range of values that satisfies an inequalityAnalyze

FormulateCombine unknowns on the left sideDetermine3(x + 2) > 5x3x + 6 > 5x3x 5x + 6 > 02x + 6 > 02x > 6 x < 3Find the solution set of the inequality 3(x + 2) > 5x {three times x plus two is greater than five x}.

First, analyze the problem. The problem presents the inequality 3(x + 2) > 5x {three times x plus two is greater than five x} and asks for an expression giving the possible solutions.

Next, formulate a plan or strategy to solve the problem. Simplify the left side. Arrange and combine the terms so there is one unknown term on the left and one constant term on the right. Divide each side by the coefficient of the variable, flipping the inequality sign if the coefficient is negative.

Next, determine the solution to the problem. Use the distributive property. Subtract 5x from both sides. Combine terms. Subtract 6 from both sides.Divide both sides by 2 and flip the inequality sign.

Now that the solution has been determined, justify it. Applying the distributive, subtraction, and division properties produced the solution x < 3 {x is less than three}.

Last, evaluate the effectiveness of the steps, and the reasonableness of the solution. Following the steps for solving multistep equations provided a straightforward solution process. The answer is reasonable because x < 3 {x is less than three} describes a range of values that satisfies an inequality. 17Multistep Inequalities Example

Ex) Find the solution set of the inequality 4x 6 < 2(x + 1). Graph the solution on a number line.Justify

EvaluateReasonable because it describes a range of values that satisfy the inequality

AnalyzeAsks for graph of solutionFormulate

Determine4x 6 < 2(x + 1)4x 6 < 2x + 22x 6 < 22x < 8x < 4

Find the solution set of the inequality 4x 6 < 2(x + 1) {four x minus six is less than two time x plus one}. Graph the solution on a number line.

First, analyze the problem. The problem presents the multistep inequality 4x 6 < 2(x + 1) {four x minus six is less than two times x plus one} and asks for a graph of the solution.

Next, formulate a plan or strategy to solve the problem. Simplify the right side of the equation. Eliminate the variable term from the right and the constant term from the left. Divide both sides by the coefficient of the variable, flipping the inequality sign if that coefficient is negative. Graph the solution on a number line.

Next, determine the solution to the problem. Use the distributive property. Subtract from both sides. Add 6 to both sides. Divide both sides by 2. Graph the solution.

Now that the solution has been determined, justify it. The process for solving a multistep equation was used to solve the inequality, resulting in the solution x < 4 {x is less than four}.

Last, evaluate the effectiveness of the steps, and the reasonableness of the solution. The sequence of steps for solving a multistep equation provided a straightforward method for solving the inequality. The answer is reasonable because it describes a range of values that satisfy the inequality.

18Learning ObjectivesUse the properties of equality to solve multistep equations of one unknownApply the process of solving multistep equations to solve multistep inequalities

Multistep EquationsYou should now be able to After this lesson, you will be able to use the properties of equality to solve multistep equations of one unknown, and apply the process of solving multistep equations to solve multistep inequalities.

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