Question

use the hanger diagram to solve for the value of x. write an equation to represent this hanger diagram.

Explanation:

Step1: Analyze the hanger diagram

Looking at the left side of the hanger, we can see that there are 4 groups, each group consisting of \( x \) and \( 4 \). The right side has a value of \( 32 \). Wait, actually, let's re - examine. Wait, the left side: let's count the number of sets. Wait, the diagram shows that there are 4 sets? Wait, no, looking at the equation structure \( \square(\square+\square)=\square \). Let's see the left - hand side of the hanger: each "unit" seems to be a combination of \( x \) and \( 4 \), and there are 4 such units? Wait, no, the number of times the \( x \) and \( 4 \) appear. Wait, the left side has 4 "x" - like triangles? Wait, no, looking at the equation boxes. The first box is a coefficient, then inside the parentheses are two terms, and the right - hand side is 32. Let's think about the hanger balance: the left side's total weight equals the right side's weight (32). Looking at the left side, we can see that there are 4 groups, each group is \( x + 4 \)? Wait, no, maybe 4 times (\( x+4 \))? Wait, no, let's check the numbers. Wait, the right side is 32. Let's see the available numbers: 4, \( x \), 32. Wait, the equation is in the form \( 4(x + 4)=32 \)? Wait, no, wait the diagram: on the left, there are 4 "x" (triangles) and 4 "4" (squares)? Wait, no, let's count the number of \( x \) and 4. Wait, the left side: how many \( x \) and how many 4? Let's see the hanger: the left side has 4 triangles (x) and 4 squares (4). Wait, no, maybe it's 4 groups, each group is \( x + 4 \), so the total is \( 4(x + 4) \), and the right side is 32. Wait, but let's check the numbers given in the options below the equation: 32, 16, 4, x, 4. Wait, maybe the coefficient is 4, inside the parentheses are \( x \) and 4, and the right side is 32. So the equation is \( 4(x + 4)=32 \). Wait, but let's confirm.

Wait, the hanger is balanced, so the left - hand side (LHS) equals the right - hand side (RHS = 32). On the LHS, we can see that there are 4 units, each unit is \( x+4 \) (since there are 4 triangles (x) and 4 squares (4), so grouped as 4 times (\( x + 4 \))). So the equation is \( 4(x + 4)=32 \).

Step2: Solve the equation \( 4(x + 4)=32 \)

First, divide both sides of the equation by 4:
\( \frac{4(x + 4)}{4}=\frac{32}{4} \)
Simplify, we get \( x + 4 = 8 \)

Then, subtract 4 from both sides:
\( x+4 - 4=8 - 4 \)
Simplify, we get \( x = 4 \)

But first, let's fill the equation. The first box (coefficient) is 4, inside the parentheses are \( x \) and 4, and the right - hand side is 32. So the equation is \( 4(x + 4)=32 \)

Answer:

The equation is \( \boldsymbol{4(x + 4)=32} \), and solving for \( x \):

Step1: Divide both sides by 4

\( x + 4=\frac{32}{4}=8 \)

Step2: Subtract 4 from both sides

\( x=8 - 4 = 4 \)
The value of \( x \) is \( \boldsymbol{4} \), and the equation is \( \boldsymbol{4(x + 4)=32} \)