#### When solving an inequality how do can you tell if a solution has no solution?

An example of this comes from my homework itself, 4(x+1) = 2x + 4. What steps do I take so I can determine if equations have no solutions whatsoever?

Anonymous

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**How to** find **the** positive integer **solutions to** [math]\frac{**x**}{y+z}+ \frac ...

**The** first thing **to do** when **you**'re looking at **any equation is to** try **and** place it in ....
**for any** particular **solution we** find, **we** just **need to check that** it doesn't **make** ...

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Mahesh Godavarti

1

Essentially, when there is no solution, it means that there is no value of x that satisfies the equation or the inequality.

Let's start with the simplest one.

x = x + 1 (does this equation have any solutions?. Obviously not, because there is no way a number is equal to itself plus one)

How can we show this? Subtract x from both sides, then we get

0 = 1 (which cannot be true)

Therefore, x = x + 1 has no solutions.

Let's take an inequality

x > x (does this inequality have any solutions? Obviously not, because a number cannot be greater than itself!)

Subtract x from both sides we get

0 > 0 (which is not true, since 0 is not greater than 0. Therefore, x > x has no solutions).

The way to show whether an equation or inequality has no solutions is to manipulate the equation or inequality till we end up with a statement that is false.

-------------------

Let's take the homework question.

4(x + 1) = 2x + 4

4x + 4 = 2x + 4

Subtract 4 from both sides

4x = 2x

Subtract 2x from both sides

2x = 0

Divide both sides by 2

x = 0.

Therefore, this equation has a solution. That is the statement becomes true when x = 0.

-----------------------

Let's take the second example.

9 - 5x + 2 = 4 - 5x

11 - 5x = 4 - 5x

Subtract 4 from both sides

7 - 5x = -5x

Add 5x to both sides

7 = 0

(This is not a true statement! There is no value of x, that will make the left hand side of the equation equal to the right hand side. Therefore, there are no solutions.)

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