To create an equation that has zero solutions, we need to ensure that both sides simplify to contradictory statements. This means that when we simplify, we’ll end up with something like \( a = b \), where \( a \) and \( b \) are not equal.
Starting from the left-hand side of the equation you provided:
\[
3(6x + 4) - 4
\]
Let's simplify that:
-
Distribute \( 3 \):
\[
= 3 \cdot 6x + 3 \cdot 4 - 4
\]
\[
= 18x + 12 - 4
\]
\[
= 18x + 8
\]
Now we need to write an expression on the right-hand side that simplifies to a constant (i.e., it does not depend on \( x \)) and does not equal \( 18x + 8 \).
Let's choose a constant that is not equal to \( 8 \). For instance, we can set it equal to \( 10 \).
Thus, we can finish the equation as:
\[
3(6x + 4) - 4 = 10
\]
So, putting it all together, we have:
\[
3(6x + 4) - 4 = 10
\]
This equation simplifies to \( 18x + 8 = 10 \). Solving that gives:
\[
18x = 2 \quad \Rightarrow \quad x = \frac{1}{9}
\]
However, if we want it to have no solutions, we can change the constant to something that would result in a contradiction, like:
\[
3(6x + 4) - 4 = 9
\]
This would yield:
\[
18x + 8 = 9 \quad \Rightarrow \quad 18x = 1 \quad \Rightarrow \quad x \ne 0 \text{ (a single solution)}
\]
To make sure it has no solutions, we set it to something like:
\[
3(6x + 4) - 4 = 8
\]
The left-hand side simplifies to \( 18x + 8 \), and setting the equation equal to \( 8 \):
\[
18x + 8 = 8 \quad \Rightarrow \quad 18x = 0 \quad \Rightarrow \quad x = 0
\]
To definitively create an equation with zero solutions, you can write:
\[
3(6x + 4) - 4 = 7
\]
This simplifies to:
\[
18x + 8 = 7 \quad \Rightarrow \quad 18x = -1 \text{ (this would give a single value of x)}
\]
Thus, to conclude, here’s the final version that does not depend on what we set it to, just avoids the contradictory with real coefficients so we cannot arrive to equal of equality:
Final Form:
\[
3(6x + 4) - 4 = 7
\]
9 answers