# Why do some equations or inequalities have no solution?

I've seen some equations and inequalities that have no solution. Examples of these are$$3m+4=3m-9$$$$128y-10\lt128y-25$$$$10t+45\ge2(5t+23)$$The third example evaluates to$$10t+45\ge10t+46$$using the Distributive Property. Maybe they have no solution because the coefficients of the variables are the same on both sides on each example, but the constants don't make sense on both sides according to the symbols in between the expressions. I hope I'm on the right track about why these have no solutions! I have a good taste in your answers!

• I don't know why this things disturbs you? There are a lot of equations and inequalities that have no solutions. – brick Oct 28 '14 at 23:06

Have a look here for Inequality Properties, it is important to grasp the concepts early for this.

Basically, you can say, if:

$10t+45\geq10t+46$

Then, by subtracting $10t$ from both sides:

$10t+45-10t\geq10t+46-10t$

We'd get:

$45\geq46$

Which, as we know, is a false statement. There is no value (or range of values) for $t$ to make the original inequality true.

• Well, 45 is never greater than or equal to 46, so it'll always be unequal. So, this inequality has no solution. – Mathster Oct 28 '14 at 23:36

You can clearly cancel out the terms that exists on both sides:

For example: $3m + 4 = 3m - 9$ is equivalent to $4 = -9$ which which is not true for all $m$.

Same goes for other equations/inequalities that you have mentioned.

• Yeah, that might be right, right? You just subtracted $3m$ from both sides, which gets rid of both of those terms and leaves you with 4=-9, which isn't true. – Mathster Oct 28 '14 at 23:10