To determine which set of numbers contains all solutions to the inequality \(4x + 7 \neq 23\), we first need to solve the equation \(4x + 7 = 23\) to identify the values of \(x\) that do not satisfy the inequality.
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Start from the equation: \[ 4x + 7 = 23 \]
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Subtract 7 from both sides: \[ 4x = 23 - 7 \] \[ 4x = 16 \]
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Divide both sides by 4: \[ x = 4 \]
The only solution to the equation \(4x + 7 = 23\) is \(x = 4\). Therefore, the inequality \(4x + 7 \neq 23\) holds for all numbers except \(x = 4\).
Now we can examine each of the provided sets of numbers to see which set does not include the number 4 (as all the numbers in that set would satisfy the inequality):
- Set 1: \(4, 5, 6, 7\) → Includes 4 (not a solution).
- Set 2: \(2, 3, 5, 6\) → Does not include 4 (all are valid solutions).
- Set 3: \(1, 2, 3, 4\) → Includes 4 (not a solution).
- Set 4: \(3, 4, 6, 7\) → Includes 4 (not a solution).
The only set that contains numbers all of which are solutions to \(4x + 7 \neq 23\) is:
2, 3, 5, 6