To solve the expression \( 6 - \frac{1}{4} \cdot 28 - 2 \frac{1}{4} _ \frac{3}{4} + 4 \cdot 2 - 11 \), we need to determine which operator (e.g., +, -, ×, ÷) can be inserted in place of the underscore to make the equation true.
Let's break down the expression step by step.
-
Calculate \( 6 - \frac{1}{4} \cdot 28 - 2 \frac{1}{4} \):
- First, calculate \( \frac{1}{4} \cdot 28 = 7 \).
- Now substitute it back: \( 6 - 7 - 2 \frac{1}{4} \).
- Convert \( 2 \frac{1}{4} \) to an improper fraction: \( 2 \frac{1}{4} = \frac{9}{4} \).
- Therefore, the expression becomes \( 6 - 7 - \frac{9}{4} \).
Now, calculate \( 6 - 7 = -1 \).
- So, we have \( -1 - \frac{9}{4} = -1 - 2.25 = -3.25 = -\frac{13}{4} \).
-
Now let's evaluate the right side \( \frac{3}{4} + 4 \cdot 2 - 11 \):
- First, calculate \( 4 \cdot 2 = 8 \).
- Next, substitute it back: \( \frac{3}{4} + 8 - 11 \).
- Now calculate \( 8 - 11 = -3 \).
- Therefore, the expression becomes \( \frac{3}{4} - 3 \), which equals \( \frac{3}{4} - \frac{12}{4} = -\frac{9}{4} \).
Now we have: \[ -\frac{13}{4} _ -\frac{9}{4} \]
Next, we can test the operators:
-
If we test with \( _ = < \): \(-\frac{13}{4} < -\frac{9}{4}\), which is true.
-
If we test with \( _ = = \): \(-\frac{13}{4} = -\frac{9}{4}\), which is false.
-
If we test with \( _ = > \): \(-\frac{13}{4} > -\frac{9}{4}\), which is false.
-
If we test with \( _ = + \): \(-\frac{13}{4} + -\frac{9}{4} = -\frac{22}{4} = -\frac{11}{2}\), which doesn't make it true.
-
If we test with \( _ = - \): \(-\frac{13}{4} - -\frac{9}{4} = -\frac{13}{4} + \frac{9}{4} = -\frac{4}{4} = -1\), which doesn't make it true.
-
If we test with \( _ = × \): \(-\frac{13}{4} \times -\frac{9}{4} = \frac{117}{16}\), which doesn't make it true.
-
If we test with \( _ = ÷ \): \(-\frac{13}{4} ÷ -\frac{9}{4} = \frac{13}{9}\), which doesn't make it true.
Only with \( < \) did we find a true statement: \[ -\frac{13}{4} < -\frac{9}{4} \] Thus, the operator that can be inserted to make the statement true is \( < \).