Question

for example, solving \\(2x + 2 + 3 = 2x + 5\\).

the equation can be simplified to ________, which is a ture mathematical statement.

in this case, the equation will always holds with any value of \\(x\\).

thus, there is(are) ________ solution(s).

for another example, solving \\(3x + 3 = 3x + 2 - 1\\).

the equation can be simplified to ________, which is a false mathematical statement.

in this case, the equation will never holds with any value of \\(x\\).

thus, there is(are) ________ solution(s).

math exploration 4

1 sat how many solutions does the equation \\(2x + 3 = 2x - 4\\) have?

Explanation:

Simplify the first equation

$$
LATEXBLOCK0
$$

Determine solutions for the first equation

Since \(5 = 5\) is always true, any real number \(x\) is a solution.

$$ \text{Number of solutions} = \text{infinitely many} $$

Simplify the second equation

$$
LATEXBLOCK1
$$

Determine solutions for the second equation

Since \(3 = 1\) is false, no value of \(x\) satisfies the equation.

$$ \text{Number of solutions} = \text{no} $$

Solve the final question

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LATEXBLOCK2
$$

Since \(3 = -4\) is false, there are no solutions.

Answer:

Question 1

For example, solving \(2x + 2 + 3 = 2x + 5\).
The equation can be simplified to <blank>\(2x + 5 = 2x + 5\)</blank> (or <blank>\(5 = 5\)</blank>), which is a true mathematical statement.
In this case, the equation will always hold with any value of \(x\).
Thus, there is(are) <blank>infinitely many</blank> solution(s).

Question 2

For another example, solving \(3x + 3 = 3x + 2 - 1\).
The equation can be simplified to <blank>\(3x + 3 = 3x + 1\)</blank> (or <blank>\(3 = 1\)</blank>), which is a false mathematical statement.
In this case, the equation will never hold with any value of \(x\).
Thus, there is(are) <blank>no</blank> solution(s).

Question 3

How many solutions does the equation \(2x + 3 = 2x - 4\) have?
<blank>no solution</blank> (or <blank>0</blank>)