To identify which equation has infinitely many solutions, we need to determine which equation is true for all values of [tex]\( x \)[/tex]. This happens when both sides of the equation simplify to the same expression. Let's solve each equation step-by-step:
### Equation A
[tex]\[ 12 + 4x = 6x + 10 - 2x \][/tex]
Simplify the right-hand side first:
[tex]\[ 6x + 10 - 2x = 4x + 10 \][/tex]
Now we have:
[tex]\[ 12 + 4x = 4x + 10 \][/tex]
Subtract [tex]\( 4x \)[/tex] from both sides:
[tex]\[ 12 = 10 \][/tex]
This statement is false, which means there are no solutions to this equation.
### Equation B
[tex]\[ 5x + 14 - 4x = 23 + x - 9 \][/tex]
Simplify both sides:
[tex]\[ 5x - 4x + 14 = x + 14 \][/tex]
Which simplifies to:
[tex]\[ x + 14 = x + 14 \][/tex]
This statement is an identity, which means it is true for all values of [tex]\( x \)[/tex]. Therefore, this equation has infinitely many solutions.
### Equation C
[tex]\[ x + 9 - 0.8x = 5.2x + 17 - 8 \][/tex]
Simplify both sides:
[tex]\[ x - 0.8x + 9 = 5.2x + 9 \][/tex]
Which simplifies to:
[tex]\[ 0.2x + 9 = 5.2x + 9 \][/tex]
Subtract [tex]\( 5.2x \)[/tex] and 9 from both sides:
[tex]\[ 0.2x - 5.2x = 0 \][/tex]
[tex]\[ -5x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
This equation has a single solution ([tex]\( x = 0 \)[/tex]).
### Equation D
[tex]\[ 4x - 2x = 20 \][/tex]
Simplify the left-hand side:
[tex]\[ 2x = 20 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 10 \][/tex]
This equation has a single solution ([tex]\( x = 10 \)[/tex]).
### Conclusion
Equation B is the one that has infinitely many solutions because it simplifies to an identity. Therefore, the answer is:
B. [tex]\( 5x + 14 - 4x = 23 + x - 9 \)[/tex]
