We are given the equation $|x+6| = 2x$ and a partially completed solution. We need to fill in the blanks numbered from 32 to 39 with the appropriate values or symbols from the provided options.

1. Problem Description

We are given the equation $|x+6| = 2x$ and a partially completed solution. We need to fill in the blanks numbered from 32 to 39 with the appropriate values or symbols from the provided options.

2. Solution Steps

(i) Case 1: $x+6 \ge 0$. This means $x \ge -6$. So, 32-1 should be $ge$. Therefore, 32 is "シ" ($\ge$). Equation (1) is $x \ge -6$.

When $x+6 \ge 0$, $|x+6| = x+6$. Thus, $x+6 = 2x$. So, 33 is $x+6$.

Then, solving $x+6 = 2x$, we get $x=6$. Thus, 34 is

6.

Equation (2) is $x=6$.

Now, we check if $x=6$ satisfies the condition $x \ge -6$. Since $6 \ge -6$, the condition is satisfied.

(ii) Case 2: $x+6 < 0$. This means $x < -6$. So, 32-2 should be $<$. Therefore, 32 is "ケ" ($<$). Equation (3) is $x < -6$.

When $x+6 < 0$, $|x+6| = -(x+6) = -x-6$. Thus, $-x-6 = 2x$. So, 36 is $-x-6$.

Then, solving $-x-6 = 2x$, we get $3x = -6$, so $x = -2$. Thus, 37 is -

2.

Equation (4) is $x=-2$.

Now, we check if $x=-2$ satisfies the condition $x < -6$. Since $-2$ is not less than $-6$, the condition is not satisfied.

Therefore, the only solution is $x=6$. Thus, 39 is

6.

Now let's go back to filling in 35 and

3

8. Since we know that the equation $x = 6$ satisfies equation (1) and equation (2), then we should select an option regarding if (1) is satisfied by (2) or not.

Since $x=6$ is a solution for $x \ge -6$, then equation (2) satisfies equation (1). Thus, 35 is "ウ".

Since $x=-2$ does *not* satisfy equation (3), then equation (4) does not satisfy equation (3). Thus, 38 is "ク".

3. Final Answer

32: シ

33: ス

34: 6

35: ウ

36: ソ

37: -2

38: ク

39: 6