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Solving Linear Equations with Parentheses;
     "No Solution" and "All Real Numbers" (page 3 of 3)

Sections: One-step equations, Multi-step equations, "No solution" and "all x" equations

• Solve 11 + 3x – 7 = 6x + 5 – 3x

First, combine like terms; then solve:

Then the "solution" is "no solution".

When you try to solve an equation, you are making the assumption that there is a solution. When you end up with nonsense (like the "4 = 5" above), this says that your initial assumption (that there was a solution) was wrong, so there is no solution. Since "4 = 5" is utterly false, and there is no value of x that could ever make it true, then this equation has no solution.

This is different from the previous problem, where there was a value of x that would work. Don't confuse these two very different situations: "the solution exists and has the value of zero" is not the same as "no solution value exists at all". Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

And don't confuse the "no solution" type of equation above with the following type:

• Solve 6x + 5 – 2x = 4 + 4x + 1

First, I'll combine like terms; then I'll solve:

Is there any value of x that would make the above statement false? Isn't 5 always going to equal 5? In fact, since there is no "x" in the solution, the value of x is irrelevant: x can be anything I want. So the solution is "all x".

This solution could also be stated as "all real numbers". And if you solve by subtracting 5 from both sides to get "4x = 4x", another trivially-true statement, or subtract 4x and 5 from both sides to get "0 = 0", the solution would still be the same: "all x".

• Solve 9 = 3(5x – 2)

First, I have to multiply through the parentheses on the right. Then I can solve:

Then the solution is  x = 1.

• Solve 6x – (3x + 8) = 16

Be careful with taking negatives through parentheses. Simplify on the left-hand side first; then solve:

Then the solution is  x = 8.

• Solve 7(5x – 2) = 6(6x – 1)

I have to be sure to take the 7 and the 6 all the way through their respective parentheses.

Then the solution is  x = –8.

For this type of problem, take your time and write out all of your steps. Don't try to do everything in your head.

• Solve 13 – (2x + 2) = 2(x + 2) + 3x

Multiply through the parentheses (a minus sign on the left, and a two on the right), combine like terms, simplify, and solve:

Then the solution is  x = 1.

You can always check your work on equation-solving problems. The point of a solution is that it's the x-value that makes the equation true. To check your answer, plug in your solution, and make sure that the equation works. For instance, the first problem I did was "x + 6 = –3", and I got a solution of
"x = –9". Here's the check:

x + 6 = –3

(–9) + 6 = –3

–3 = –3

So the solution "checks".

This ability to check your answers can come in handy on tests. Once you've completed all the questions, go back and plug in your solutions. If the solution "checks", then you know you got that question right. If it doesn't, you have the chance to correct your mistake before you hand in the test!

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