Verified Answer :

  • We evaluate each number from the set \(\{1, -5, -7, 2, -10, -4\}\) against the inequality \(x \leq 1\).
  • We confirm that \(1, -5, -7, -10,\) and \(-4\) are less than or equal to 1, while \(2\) is greater than 1.
  • The values that satisfy the inequality are \(\boxed{1, -5, -7, -10, -4}\).

Explanation

  1. Problem Analysis

We are given the inequality \(x \leq 1\) and a set of candidate numbers: \(\{1, -5, -7, 2, -10, -4\}\). To solve this, we must test each number individually to see if it makes the inequality true. A number \(n\) satisfies the inequality if \(n\) is either equal to 1 or less than 1.

  1. Testing Candidates

We now evaluate each candidate value against the inequality \(x \leq 1\):

  1. For \(x = 1\): \(1 \leq 1\) is true.
  2. For \(x = -5\): \(-5 \leq 1\) is true, because any negative number is less than 1.
  3. For \(x = -7\): \(-7 \leq 1\) is true, because any negative number is less than 1.
  4. For \(x = 2\): \(2 \leq 1\) is false, because 2 is greater than 1.
  5. For \(x = -10\): \(-10 \leq 1\) is true, because any negative number is less than 1.
  6. For \(x = -4\): \(-4 \leq 1\) is true, because any negative number is less than 1.
  1. Conclusion

After checking each value, we find that the values \(1, -5, -7, -10,\) and \(-4\) all satisfy the condition \(x \leq 1\). The value \(2\) does not satisfy the condition. Therefore, the list of solutions is \(\{1, -5, -7, -10, -4\}\).

Examples

In real life, this type of inequality often represents a limit or a threshold. For instance, if a recipe requires an oven temperature to be at most \(1^\circ \text{C}\) (or some specific threshold) to prevent an ingredient from spoiling, you would identify all safe temperatures. Here, if the threshold is 1, any value less than or equal to that is acceptable, ensuring your project remains within safe operational limits.