Question

which of these equations has no solutions?
4(x + 3) = 4(x - 3)
3x + x - 3 = \frac{1}{3}(12x - 9)
2x + 3 - 4x = 4x + 3
which statement explains a way you can tell the equation has no solutions?
it is equivalent to an equation that has the same variable terms but different constant terms on each side of the equal sign.
it is equivalent to an equation that has the same variable terms and the same constant terms on each side of the equal sign.
it is equivalent to an equation that has different variable terms on each side of the equation.

Explanation:

Part 1: Finding the equation with no solutions

Equation 1: \( 4(x + 3) = 4(x - 3) \)

Step 1: Expand both sides

Left side: \( 4(x + 3)=4x + 12 \)
Right side: \( 4(x - 3)=4x - 12 \)

Step 2: Subtract \( 4x \) from both sides

\( 4x + 12-4x=4x - 12-4x \)
Simplify: \( 12=- 12 \)
This is a false statement, so this equation has no solutions.

Equation 2: \( 3x + x - 3=\frac{1}{3}(12x - 9) \)

Step 1: Simplify left and right sides

Left side: \( 3x+x - 3 = 4x-3 \)
Right side: \( \frac{1}{3}(12x - 9)=4x - 3 \)

Step 2: Subtract \( 4x \) and add 3 to both sides

\( 4x-3-4x + 3=4x - 3-4x + 3 \)
Simplify: \( 0 = 0 \)
This is a true statement, so the equation has infinitely many solutions.

Equation 3: \( 2x + 3-4x=4x + 3 \)

Step 1: Simplify left side

\( 2x-4x + 3=-2x + 3 \)

Step 2: Subtract \( 4x \) and 3 from both sides

\( -2x+3-4x - 3=4x + 3-4x - 3 \)
Simplify: \( -6x=0 \)

Step 3: Solve for \( x \)

\( x = 0 \)
This equation has one solution.

Part 2: Explaining how to tell an equation has no solutions

An equation has no solutions when, after simplifying, we get an equation where the variable terms are the same (so they cancel out) but the constant terms are different. For example, in \( 4(x + 3)=4(x - 3) \), we ended up with \( 12=-12 \), where the \( 4x \) terms canceled (same variable terms) and the constants (12 and - 12) were different. If the variable terms are the same and constants are different, the equation is a contradiction and has no solutions. If variable terms are same and constants are same, it has infinitely many solutions. If variable terms are different, we can solve for the variable.

Answer:

It is equivalent to an equation that has the same variable terms but different constant terms on each side of the equal sign.

Final Answers:

• The equation with no solutions is \( \boldsymbol{4(x + 3)=4(x - 3)} \)
• The statement that explains how to tell an equation has no solutions is "It is equivalent to an equation that has the same variable terms but different constant terms on each side of the equal sign."