The document discusses whether solutions exist for various equations. In three sentences: It asks if solutions exist for different equations and provides explanations for each. For the equations 3x = 6, x^2 = 9, and x^2 >= 0 + 9, solutions do exist as you can find values of x that satisfy the equations. However, for the equations 3x - 5 = 3x + 6, x+4 = x+3, and the fraction equation given, no solutions exist as the equations cannot be satisfied no matter the value of x.

1. Does a solution exists
for each equation?
Explain!
without actually solving the equation

3. Yes
If you add six to five you get 11, and there
exists a value of x such that 3x = 6

5. Yes
Adding nine to one gives you 10,
and there exists a value of x such that x2 = 9.

7. No
For any value of x,
x2 will be greater than or equal to zero.
If we add a non-negative number to nine,
it is impossible to end up with one.

9. No
No matter what x is, 3x is the same on both sides.
Thus the number 3x minus 5 cannot be the same as
the number 3x plus 6.

11. No
A fraction is equal to one
when the numerator and denominator
are equal to each other (and not zero);
x+4 and x+3 can never be equal
to each other no matter the value of x.

13. No
A fraction is equal to one when the numerator and
denominator are equal to each other (and not zero).
If the top equation = 2(x-1),
the numerator and denominator would be equal.
With the numerator only x - 1, the denominator is always
twice the value of the numerator no matter the value of x.