Question

Find the missing number so that the equation has no solutions.
–
4x–2=
x–11–9x

Answer 1

To find the missing number so that the equation has no solutions, we start with the given equation:

\[
-4x - 2 = x - 11 - 9x
\]

First, simplify the right side:

\[
x - 11 - 9x = -8x - 11
\]

Now, the equation becomes:

\[
-4x - 2 = -8x - 11
\]

Next, let's rearrange the terms. We can isolate \(x\) on one side by adding \(8x\) to both sides:

\[
-4x + 8x - 2 = -11
\]

This simplifies to:

\[
4x - 2 = -11
\]

Now add 2 to both sides:

\[
4x = -11 + 2
\]

\[
4x = -9
\]

Now divide both sides by 4:

\[
x = -\frac{9}{4}
\]

In order for the original equation \( -4x - 2 = x - 11 - 9x \) to have no solutions, we would need a scenario where both sides are equal but yield a contradiction (e.g., something like \(0 = 5\)).

We can create such a case by manipulating the constants in the equation.

If we consider modifying the constant on the left side, for example, to match a case where both sides cannot equal when both sides are simplified to \(Ax + B = Cx + D\), we can set it such that the coefficients of \(x\) are equal but the constants lead to a contradiction.

Let’s set the original left (–4x) side equal to the negative of the simplification of the right side when we create it:

Let's take the calculated form:

Original combined form rearranged gives us:

If we make the right side equal to \(0\) upon transformation, like letting the constants define a contradiction by the following modified left expanding:
Let’s say we have:

-4x - 2 could correspond to becoming a contradiction such as changing it to either x-12=0 when simplified like above but noticing the contradiction effect:

Now we can set it to have:

Thus on one side cause the contradictions by whimsically setting the value so say modifying:

\[
-4x - a = -8x + b \text{(where b was previously manipulated to equal something wrong)}
\]

If we set \(b\) to thus not equate by equating towards a constant that doesn't resolve to balance like say any correlations of constants relating (rather than x) altered to identities that don't equal:

Thus it leads to phrased answers involving miscalculations or errors embedded:

The values needed for lack of solution further signifies; thus, 4 being a suitably mismatched constant unified could have effectively generated towards equating identities that inherent towards semblance of solutions but fail.

One form achieving toward false identity could be formed thus is:

Hence, the missing number in terms of constraining formulation leads to \(12\) as it would define:
\[
0 = 12 \text{ would yield no solutions hence finalize here.}
\]

Thus, marking effectively towards the idea having formed contradictions regardless summed constants but poignantly favor meaning towards equating would yield:

**The missing number, therefore, is \(12\)** to flat out produce an unsolvable equation; thus confirming overlaid contradictions involved.