26 + 2 x 8 - 4 / 1 = 14

To solve this problem, we need to find which two signs, when interchanged in the given equation, make the equation correct. The equation is:

$$26 + 2 \times 8 - 4 \div 1 = 14$$

First, let's evaluate the original expression to see its current value:

$$26 + 2 \times 8 - 4 \div 1$$

Following the order of operations (BODMAS/PEMDAS - Brackets, Orders, Division/Multiplication, Addition/Subtraction):

1. Multiplication: $2 \times 8 = 16$
2. Division: $4 \div 1 = 4$ Substitute these values back into the expression:

$$26 + 16 - 4$$

3. Addition: $26 + 16 = 42$
4. Subtraction: $42 - 4 = 38$ The current value of the left side is 38, which is not equal to 14. So, we need to interchange signs.

Now, let's test each option:

• Option 1: Interchange $-$ and $\times$ The equation becomes:

  $$26 + 2 - 8 \times 4 \div 1 = 14$$

  Evaluate the left side:

  1. Multiplication: $8 \times 4 = 32$
  2. Division: $32 \div 1 = 32$ Substitute these values:
  $$26 + 2 - 32$$
  3. Addition: $26 + 2 = 28$
  4. Subtraction: $28 - 32 = -4$ Since $-4 \neq 14$, this option is incorrect.

• Option 2: Interchange $=$ and $-$ This means the original '-' sign becomes '=' and the original '=' sign becomes '-'. The equation would be:

  $$26 + 2 \times 8 = 4 - 1$$

  Evaluate the left side:

  1. Multiplication: $2 \times 8 = 16$
  2. Addition: $26 + 16 = 42$ Evaluate the right side:
  3. Subtraction: $4 - 1 = 3$ Since $42 \neq 3$, this option is incorrect.

• Option 3: Interchange $=$ and $\times$ This means the original '×' sign becomes '=' and the original '=' sign becomes '×'. The equation would be:

  $$26 + 2 = 8 \times 4 \div 1$$

  Evaluate the left side:

  1. Addition: $26 + 2 = 28$ Evaluate the right side:
  2. Multiplication: $8 \times 4 = 32$
  3. Division: $32 \div 1 = 32$ Since $28 \neq 32$, this option is incorrect.

• Option 4: Interchange $+$ and $-$ The equation becomes:

  $$26 - 2 \times 8 + 4 \div 1 = 14$$

  Evaluate the left side:

  1. Multiplication: $2 \times 8 = 16$
  2. Division: $4 \div 1 = 4$ Substitute these values:
  $$26 - 16 + 4$$
  3. Subtraction: $26 - 16 = 10$
  4. Addition: $10 + 4 = 14$ Since $14 = 14$, this option makes the equation correct.

Therefore, interchanging the $+$ and $-$ signs makes the equation correct.

The final answer is $\text{4}$.