which symbol in place of the question mark makes the following inequality true for the solutions 4,5,6?

Question

4x ? 16
given:
greater than or equal to
less than or equal to
equal
greater than

Answer 1

To determine which symbol makes the inequality \(4x ? 16\) true for the solutions \(x = 4\), \(x = 5\), and \(x = 6\), we will test each symbol one by one.

1. Greater than or equal to (≥): \[ 4x \geq 16 \implies x \geq 4 \] For \(x = 4\): \(4 \times 4 = 16\) (True)
   For \(x = 5\): \(4 \times 5 = 20\) (True)
   For \(x = 6\): \(4 \times 6 = 24\) (True)
   This works for all three values.

2. Less than or equal to (≤): \[ 4x \leq 16 \implies x \leq 4 \] For \(x = 4\): \(4 \times 4 = 16\) (True)
   For \(x = 5\): \(4 \times 5 = 20\) (False)
   This does not work for \(x = 5\).

3. Equal (=): \[ 4x = 16 \implies x = 4 \] For \(x = 4\): \(4 \times 4 = 16\) (True)
   For \(x = 5\): \(4 \times 5 = 20\) (False)
   This does not work for \(x = 5\) or \(x = 6\).

4. Greater than (>): \[ 4x > 16 \implies x > 4 \] For \(x = 4\): \(4 \times 4 = 16\) (False)
   For \(x = 5\): \(4 \times 5 = 20\) (True)
   For \(x = 6\): \(4 \times 6 = 24\) (True)
   This does not work for \(x = 4\).

The only symbol that makes the inequality \(4x ? 16\) true for all three values \(x = 4\), \(x = 5\), and \(x = 6\) is greater than or equal to (≥).