An example of something that doesn't have a solution is something like x+2 = x+3
If we subtract x from both sides, then we end up with 2 = 3, which is always false.
No matter what we plug in for x, the original equation will always be false. The right hand side is always 1 larger than the left side. So that's why we don't have any solutions here.
Side note: equations of this form are known as contradictions (or we could say the equation is inconsistent).
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An example of something that has one solution is 3x+2 = 2x+7
Solving this equation leads us to...
3x+2 = 2x+7
3x-2x = 7-2
1x = 5
x = 5
To verify the solution, we plug it back into the original equation
3x+2 = 2x+7
3(5)+2 = 2(5)+7
15+2 = 10+7
17 = 17
We get the same thing on both sides, so we get a true statement. This confirms that x = 5 is the solution to 3x+2 = 2x+7.
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An example of an equation with infinitely many solutions is 2x+4 = 2(x+2)
Notice how both sides are the same thing. The 2(x+2) distributes out to get 2x+4
Since we have the exact same identical expression on both sides, this ultimately means no matter what we plug in for x, we'll get a true statement. True statements (like the conclusion at the last section) are simply anything with the same number on both sides after simplifying everything.
Side note: equations of this form are known as identities
