Question

Enter the number that makes the algebraic expression equivalent expression 1: 4x + 5 (x - 2) + blank expression 2: blank x - 11 - x + 9

Answer 1

To solve the problem and find a number that makes the two algebraic expressions equivalent, we need to simplify both expressions and set them equal to each other.

**Expression 1:**
\[ 4x + 5(x - 2) + \text{blank} \]

Let's simplify expression 1:
1. Distribute \(5\) in the term \(5(x - 2)\):
\[
5(x - 2) = 5x - 10
\]
2. Substitute this back into original expression:
\[
4x + 5x - 10 + \text{blank}
\]
3. Combine like terms:
\[
(4x + 5x) - 10 + \text{blank} = 9x - 10 + \text{blank}
\]

So, expression 1 simplifies to:
\[ 9x - 10 + \text{blank} \]

**Expression 2:**
\[ \text{blank} \, x - 11 - x + 9 \]

Let's simplify expression 2:
1. Combine the constant terms \(-11\) and \(9\):
\[
-11 + 9 = -2
\]
2. Now rewrite it:
\[
\text{blank} \, x - x - 2
\]
3. Combine the \(x\) terms:
\[
(\text{blank} - 1)x - 2
\]

So, expression 2 simplifies to:
\[ (\text{blank} - 1)x - 2 \]

**Set the two expressions equal to each other:**
\[
9x - 10 + \text{blank} = (\text{blank} - 1)x - 2
\]

Now, let's compare the coefficients of \(x\) and the constant terms:

1. For the \(x\) coefficients:
\[
9 = \text{blank} - 1
\]
Solving for the blank:
\[
\text{blank} = 10
\]

2. For the constant terms (now substituting the blank):
\[
-10 + 10 = -2
\]
Which simplifies to:
\[
0 = -2 \quad (\text{this will not hold})
\]

We only need to satisfy the \(x\) coefficient since the blank value must comply with the equation. Therefore, the final value of blank we determined is:
\[
\text{blank} = 10
\]

Thus, the number that makes the algebraic expression equivalent is:
\[
\boxed{10}
\]