Clear the x terms, then read what is left
To solve an equation with x on both sides, you take the smaller x term off both sides. Usually some x is left over, and the equation then has exactly one solution. But sometimes the x terms are equal, so taking one off both sides removes x completely. What is left is a statement with no x in it, and that statement decides how many solutions there are.
There are only three possible answers: no solution, exactly one solution, or infinitely many solutions.
No solution
Solve 2x + 1 = 2x + 5. Subtract 2x from both sides, and the equation becomes 1 = 5.
1 = 5 is false, and there is no x left in it to change. Whatever number x is, the left side is 4 less than the right side. So no value of x makes the two sides equal, and the equation has no solution.
Draw each side as a line on a grid. As x goes up by 1, both 2x + 1 and 2x + 5 go up by 2, so the two lines are parallel. The line 2x + 5 is always 4 above the line 2x + 1, and parallel lines never meet.
The two lines climb at the same rate and stay 4 apart, so they never meet: no solution.
Infinitely many solutions
Solve 2x + 4 = 2(x + 2). Expand the bracket on the right: 2x + 4 = 2x + 4. The two sides are now exactly the same expression. Subtract 2x from both sides, and the equation becomes 4 = 4.
4 = 4 is true, and there is no x in it. So it is true whatever x is: every number is a solution, and there are infinitely many solutions. An equation that is true for every value of x is called an identity.
Drawn on a grid, both sides give the same line. Every point on that line is a place where the two sides agree.
2x + 4 and 2(x + 2) draw the same line, so the two sides agree at every x.
Exactly one solution
Solve 2x + 1 = 7 − x. The x terms are different, so clearing them leaves some x behind. Add x to both sides: 3x + 1 = 7. Subtract 1: 3x = 6. Divide by 3: x = 2.
An x survived the clearing, so it can be solved for, and it has exactly one value. Check: 2 × 2 + 1 = 5 and 7 − 2 = 5.
On a grid, 2x + 1 goes up as x increases and 7 − x goes down. Lines with different steepness cross at exactly one point, and the x at that crossing is the solution.
The lines cross once, at x = 2, where both sides are 5.
Deciding without drawing
You do not need a graph to decide. Simplify both sides, then look at the x terms first.
If the two sides have different coefficients of x, there is exactly one solution. If the coefficients are the same, the x terms cancel, and the numbers left decide: different numbers, such as 1 = 5, mean no solution; equal numbers, such as 4 = 4, mean infinitely many.
For example, 3(x + 2) = 3x + 5 expands to 3x + 6 = 3x + 5. The x terms are both 3x, and 6 is not 5, so there is no solution.
The usual mistakes
Writing x = 0 when the x terms cancel. After 2x + 1 = 2x + 5 becomes 1 = 5, x has disappeared; nothing says x is 0. Try x = 0 in the original: 1 = 5, which is false.
Reading 4 = 4 as x = 4. The statement 4 = 4 does not mention x. It is true for every x, so the answer is every number, not just 4.