Combine Terms that contain the exact same variables increased to the very same powers. Because that example, 3x and −8x are choose terms, as space 8xy2 and 0.5xy2.

You are watching: An equation that is not true for even one real number is called a/an


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on both political parties of the equation.

Isolate the x term by subtracting 2x native both sides.


This is not a solution! you did not discover a value for x. Addressing for x the method you know how, you arrive at the false statement 4 = 5. For sure 4 can not be equal to 5!

This may make sense as soon as you consider the second line in the solution where prefer terms were combined. If you main point a number by 2 and include 4 you would certainly never obtain the same answer as once you main point that exact same number by 2 and include 5. Because there is no value of x the will ever before make this a true statement, the equipment to the equation over is “no solution”.

Be cautious that you perform not confuse the solution x = 0 with “no solution”. The systems x = 0 means that the value 0 satisfies the equation, so there is a solution. “No solution” method that there is no value, not even 0, i beg your pardon would accomplish the equation.

Also, be mindful not to do the failure of thinking that the equation 4 = 5 way that 4 and also 5 are values for x that space solutions. If girlfriend substitute this values right into the initial equation, you’ll check out that they perform not meet the equation. This is because there is important no solution—there room no worths for x that will certainly make the equation 12 + 2x – 8 = 7x + 5 – 5x true.


Example

Problem

Solve because that x.

3x + 8 = 3(x + 2)

*

Apply the distributive home to simplify.

Isolate the variable term. Due to the fact that you recognize that 8 = 6 is false, there is no solution.

Answer

There is no solution.


Advanced Example

Problem

Solve because that y.

8y = 2<3(y + 4) + y>

*

Apply the distributive residential or commercial property to simplify. When two sets of group symbols are used, advice the inner collection and climate evaluate the outer set.

 

*

Isolate the variable term by subtracting 8y native both sides of the equation. Since you understand that 0 = 24 is false, over there is no solution.

Answer

There is no solution.


Algebraic Equations through an Infinite variety of Solutions


You have actually seen the if one equation has no solution, you finish up through a false statement instead of a worth for x. You deserve to probably guess the there can be a means you might end up v a true statement instead of a value for x.


Example

Problem

Solve because that x.

5x + 3 – 4x = 3 + x

*

Combine choose terms on both sides of the equation.

Isolate the x hatchet by individually x from both sides.


You arrive at the true statement “3 = 3”. When you finish up v a true statement favor this, it means that the solution to the equation is “all actual numbers”. Try substituting x = 0 into the initial equation—you will get a true statement! shot

*
, and it likewise will check!

This equation happens to have actually an infinite variety of solutions. Any value for x the you can think of will make this equation true. As soon as you think around the context of the problem, this provides sense—the equation x + 3 = 3 + x way “some number plus 3 is same to 3 add to that same number.” We understand that this is constantly true—it’s the commutative building of addition!


Example

Problem

Solve because that x.

5(x – 7) + 42 = 3x + 7 + 2x

*

Apply the distributive property and combine favor terms to simplify.

Isolate the x ax by subtracting 5x indigenous both sides. You gain the true declare 7 = 7, so you recognize that x can be all genuine numbers.

Answer

x = all real numbers


When resolving an equation, multiplying both sides of the equation by zero is not a an excellent choice. Multiply both side of one equation through 0 will always result in one equation of 0 = 0, but an equation of 0 = 0 go not aid you recognize what the systems to the initial equation is.


Example

Problem

Solve for x.

x = x + 2

*

Multiply both political parties by zero.

While that is true the 0 = 0, and also you may be tempted to conclude that x is true the all real numbers, that is not the case.

Check:

*

Better Method:

*

For example, check and also see if x = 3 will deal with the equation.

Clearly 3 never equates to 5, for this reason x = 3 is not a solution. The equation has no solutions.

It was not helpful to have multiplied both sides of the equation by zero.

It would have been far better to have started by subtracting x native both sides, resulting in 0 = 2, leading to a false statement informing us that there are no solutions.

Answer

There is no solution.


In solving the algebraic equation 2(x – 5) = 2x + 10, you end up v −10 = 10. What walk this mean?

A) x = −10 and 10

B) there is no equipment to the equation.

C) friend must have actually made a failure in solving the equation.

D) x = all actual numbers


Show/Hide Answer

A) x = −10 and also 10

Incorrect. Any solution to an equation must satisfy the equation. If you substitute −10 right into the initial equation, you gain −30 = −10. If you instead of 10 for x in the original equation, you gain 10 = 30. The exactly answer is: over there is no equipment to the equation.

B) over there is no solution to the equation.

Correct. Anytime you end up v a false statement choose −10 = 10 it method there is no equipment to the equation.

C) friend must have made a failure in addressing the equation.

Incorrect. A false statement favor this looks choose a mistake and it’s always great to examine the answer. In this case, though, over there is no a wrong in the algebra. The exactly answer is: over there is no equipment to the equation.

D) x = all real numbers

Incorrect. If you substitute some genuine numbers right into the equation, you will see that they do not meet the equation. The exactly answer is: there is no equipment to the equation.

Advanced Question

How countless solutions space there for the equation:

*

A) there is one solution.

B) There space two solutions.

C) There room an infinite variety of solutions.

D) There are no solutions.


Show/Hide Answer

A) there is one solution.

Incorrect. Try substituting any value in because that y in this equation and think about what friend find. The correct answer is: There space an infinite number of solutions come the equation.

B) There are two solutions.

Incorrect. Shot substituting any type of two values in for y in this equation and think about what friend find. When handling sets that parentheses, make certain to advice the inner parentheses first, and then move to the external set. The exactly answer is: There space an infinite number of solutions to the equation.

C) There room an infinite number of solutions.

Correct. Once you advice the expression on either side of the equates to sign, you obtain

*
. If you to be to relocate the variables come the left side and also the constants come the right, girlfriend would end up through 0 = 0. Since you have a true statement, the equation is true because that all values of y.

D) There space no solutions.

Incorrect. Recall that statements such as 3 = 5 space indicative of one equation having no solutions. The exactly answer is: There are an infinite variety of solutions come the equation.

Application Problems


The strength of algebra is how it can assist you model real cases in order come answer questions about them. This requires you to have the ability to translate real-world problems into the language the algebra, and then have the ability to interpret the outcomes correctly. Let’s start by trying out a an easy word problem that supplies algebra because that its solution.

Amanda’s dad is twice as old together she is today. The sum of their ages is 66. Usage an algebraic equation to find the ages of Amanda and her dad.

One method to settle this difficulty is to usage trial and also error—you can pick some numbers for Amanda’s age, calculation her father’s period (which is twice Amanda’s age), and then integrate them to view if they work-related in the equation. Because that example, if Amanda is 20, then she father would be 40 because he is double as old as she is, but then their linked age is 60, not 66. What if she is 12? 15? 20? together you can see, picking random numbers is a an extremely inefficient strategy!

You have the right to represent this case algebraically, which provides another method to find the answer.


Example

Problem

Amanda’s dad is twice as old together she is today. The sum of their periods is 66. Uncover the periods of Amanda and also her dad.

We require to discover Amanda’s age and also her father’s age.

What is the trouble asking?

*

Assign a variable to the unknown. The father’s age is 2 times Amanda’s age.

*

Amanda’s age added to she father’s age is equal to 66.

*

Solve the equation for the variable.

*

Use Amanda’s period to find her father’s age.

*

Do the answers make sense?

Answer

Amanda is 22 years old, and also her dad is 44 year old.


Let’s try a new problem. Consider that the rental fee because that a landscaping an equipment includes a one-time fee plus an hourly fee. You can use algebra to create an expression that helps you determine the total cost for a range of rental situations. One equation comprise this expression would certainly be valuable for do the efforts to continue to be within a fixed price budget.


Example

Problem

A landscaper wants to rental a tree stump grinder come prepare one area for a garden. The rental company charges a $26 one-time rental fees plus $48 because that each hour the device is rented.

Write an expression for the rental expense for any number of hours.

The problem asks because that an algebraic expression because that the rental cost of the stump grinder for any number of hours. One expression will have actually terms, one of which will certainly contain a variable, yet it will certainly not contain an equal sign.

What is the problem asking?

Look in ~ the worths in the problem:

$26 = one-time fee

$48 = per-hour fee

Think around what this means, and try to identify a pattern.

1 hr rental: $26 + $48

2 hr rental: $26 + $48 + $48

3 hr rental: $26 + $48 + $48 + $48

Notice that the variety of “+ $48” in the difficulty is the exact same as the number of hours the an equipment is being rented. Since multiplication is recurring addition, friend could likewise represent it choose this:

1 hr rental: $26 + $48(1)

2 hr rental: $26 + $48(2)

3 hr rental: $26 + $48(3)

What information is essential to recognize an answer?

Now let’s usage a variable, h, to represent the variety of hours the maker is rented.

Rental for h hours: 26 + 48h

What is the variable?

What expression models this situation?

The full rental fees is determined by multiplying the number of hours through $48 and including $26.

Answer

The rental price for h hours is 26 + 48h.


Using the information provided in the problem, you were able to produce a general expression for this relationship. This way that girlfriend can discover the rental cost of the device for any variety of hours!

Let’s usage this new expression come solve an additional problem.


Example

Problem

A landscaper desires to rental a tree stump grinder to prepare one area for a garden. The rental company charges a $26 one-time rental dues plus $48 because that each hour the maker is rented.

What is the maximum number of hours the landscaper can rent the tree stump grinder, if he have the right to spend no much more than $290? (The maker cannot be rented for part of one hour.)

26 + 48h, wherein h = the number of hours.

What expression models this situation?

*

Write one equation to assist you uncover out when the expense equals $290.

Solve the equation.

*

Check the solution.

*

Interpret the answer.

Answer

The landscaper can rent the maker for 5 hours.


It is often beneficial to follow a perform of procedures to organize and also solve applications problems.

Solving applications Problems

Follow these actions to interpret problem cases into algebraic equations you can solve.

1. Read and understand the problem.

2. Identify the constants and also variables in the problem.

3. Compose an equation to stand for the problem.

4. Resolve the equation.

5. Examine your answer.

6. Write a sentence that answers the question in the applications problem.

Let’s shot applying the problem-solving actions with some new examples.


Example

Problem

Gina has discovered a good price on document towels. She wants to stock up on this for she cleaning business. Record towels price $1.25 every package. If she has actually $60 to spend, how countless packages of paper towels have the right to she purchase? compose an equation the Gina can use to resolve this problem and show the solution.

The trouble asks for how plenty of packages of paper towels Gina can purchase.

What is the difficulty asking you?

The document towels expense $1.25 per package.

Gina has actually $60 to invest on document towels.

What room the constants?

Let ns = the number of packages of file towels.

What is the variable?

*

What equation represents this situation?

Solve because that p.

Divide both sides of the equation through 1.25

60 ÷ 1.25 = 6,000 ÷ 125

5 00

1,000

1,000

0

*

Check your solution. Instead of 48 in for p in your equation.

Answer

Gina deserve to purchase 48 packages of document towels.


Example

Problem

Levon and Maria to be shopping because that candles to decorate tables at a restaurant. Levon to buy 5 packages of candles plus 3 solitary candles. Maria bought 11 single candles add to 4 packages of candles. Each package that candles has the same number of candles. After finishing shopping, Maria and also Levon realized that they had actually each purchase the exact same exact number of candles. How plenty of candles space in a package?

The trouble asks how numerous candles are included in one package.

What is the trouble asking you?

Levon to buy 5 packages and also 3 single candles.

Maria purchase 4 packages and 11 single candles.

What space the constants?

Let c = the number of candles in one package.

What is the variable?

*

What expression to represent the number of candles Levon purchased?

What expression to represent the variety of candles Maria purchased?

What equation represents the situation?

Maria and also Levon purchase the same variety of candles.

*
 

Solve because that c.

Subtract 4c from both sides.

Subtract 3 native both sides.

*

Check her solution.

Substitute 8 because that c in the initial equation.

Answer

There space 8 candles in one parcel of candles.


Advanced Example

Problem

The money from 2 vending machines is gift collected. One maker contains 30 dissension bills and also a bunch the dimes. The other machine contains 38 dollar bills and also a bunch of nickels. The variety of coins in both devices is equal, and also the quantity of money that the machines accumulated is likewise equal. How many coins are in each machine?

The difficulty asks how plenty of coins room in each machine.

What is the difficulty asking you?

One device has 30 dissension bills and also a bunch that dimes.

Another maker has 38 dissension bills and also a bunch the nickels—the same variety of coins as the first machine.

What space the constants and what space the unknowns?

Let c = the variety of coins in every machine.

What is the variable?

*

What expression represents the amount of money in the first machine?

*

What expression represents the lot of money in the 2nd machine?

*

What equation represents the situation?

The amount of money in both devices is the same.

*

Solve for c.

*

Check her solution.

Substitute 160 because that c in the initial equation.

Answer

There room 160 coins in each machine.


Advanced Question

Albert and Bryn are buying liquid at the corner store. Albert buys 5 bags and also 3 individual pieces; Bryn buys 3 bags and also then eats 2 piece of liquid from among the bags. Each bag has actually the same number of pieces of candy.

After Bryn eats the 2 pieces, she has actually exactly fifty percent the variety of pieces the candy together Albert. How plenty of pieces that candy space in each bag?

Pick the equation that could be used to solve the problem above. Usage the change b to represent the variety of pieces of candy in one bag.

A)

B)

C)

D)


Show/Hide Answer

A)

Incorrect. For the equation  to be correct, the case would need to be “Albert had 5 bags and 3 pieces, and also Bryn had actually 3 bags and also 2 pieces. Albert had half as much as Bryn.” The correct answer is: .

B)

Incorrect. Because that the equation  to it is in correct, the instance would have to be “Albert had 5 bags and also 3 pieces, and Bryn had 3 bags and also 2 pieces. They had the very same amount.” The exactly answer is: .

C)

Correct. The lot of candy the Albert has can be stood for by 5b + 3, and the amount of candy Bryn has can be represented by 3b – 2. Since Bryn has half as lot as Albert, the final equation is .

D)

Incorrect. For the equation  to be correct, the instance would have to be “Albert had actually 3 bags and 3 pieces, and also Bryn had 5 bags and then ate 2 pieces. Bryn had half as much as Albert.” The correct answer is .

See more: How Many Electrons Are In A Single Covalent Bond ? Covalent Bonds


Summary


Some equations are taken into consideration special cases. These are equations that have actually no solution and equations whose systems is the set of all genuine numbers. When you use the steps for solving an equation, and you acquire a false statement quite than a value for the variable, over there is no solution. Once you use the procedures for fixing an equation, have avoided multiplying both political parties of the equation by zero, and also you gain a true statement rather than a worth for the variable, the systems is all genuine numbers. Algebra is a powerful tool for modeling and also solving real-world problems.