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
")">like terms 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 realworld 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 workrelated 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.
Let’s try a new problem. Consider that the rental fee because that a landscaping an equipment includes a onetime 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.
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.
It is often beneficial to follow a perform of procedures to organize and also solve applications problems.
Let’s shot applying the problemsolving actions with some new examples.
