Solve. –2∣x – 5∣ < –10
(1 point)
Responses
x ≤ –5 or x ≥ 5
x ≤ –5 or x ≥ 5
–5 ≤ x ≤ 5
–5 ≤ x ≤ 5
x ≤ 0 or x ≥ 10
x ≤ 0 or x ≥ 10
–10 ≤ x ≤ 10
–10 ≤ x ≤ 10
explain why
To solve the inequality –2∣x – 5∣ < –10, we can start by isolating the absolute value expression. We can do this by dividing both sides of the inequality by -2 (since -2 is negative, we need to reverse the inequality sign).
Dividing both sides by -2, we get:
∣x – 5∣ > 5
Next, we can consider two cases for x – 5: x – 5 ≥ 0 and x – 5 < 0.
Case 1: x – 5 ≥ 0
In this case, the absolute value expression is positive, so we can remove the absolute value sign.
x – 5 > 5
Adding 5 to both sides, we get:
x > 10
Case 2: x – 5 < 0
In this case, the absolute value expression is negative, so we need to reverse the inequality sign and remove the absolute value sign.
–(x – 5) > 5
Expanding the negative sign, we get:
- x + 5 > 5
Subtracting 5 from both sides, we get:
- x > 0
Multiplying both sides by -1 (and reversing the inequality sign since we are multiplying by a negative number), we get:
x < 0
Combining the solutions from both cases, we have:
x < 0 or x > 10
However, we need to consider the original inequality where the absolute value expression is less than -10.
Since the absolute value of any number is nonnegative, it can never be less than a negative number. Therefore, there are no solutions in this case.
Thus, the overall solution is:
x < 0 or x > 10
However, if we examine the given answer choices, we see that none of the choices match our solution. Therefore, it seems that there may be an error in the provided answer choices. The correct solution would be x < 0 or x > 10, as explained above.
what is the question is -2∣x-5∣\le-10?
The question is -2∣x-5∣ ≤ -10.
Solve. -2∣x-5∣ ≤ -10
answers
x ≤ –5 or x ≥ 5
x ≤ –5 or x ≥ 5
–5 ≤ x ≤ 5
–5 ≤ x ≤ 5
x ≤ 0 or x ≥ 10
x ≤ 0 or x ≥ 10
–10 ≤ x ≤ 10
The correct answer is -5 ≤ x ≤ 5.
To solve the inequality -2∣x-5∣ ≤ -10, we can start by dividing both sides of the inequality by -2 (since -2 is negative, we need to reverse the inequality sign).
Dividing both sides by -2, we get:
∣x-5∣ ≥ 5
Next, we can consider two cases for x - 5: x - 5 ≥ 0 and x - 5 < 0.
Case 1: x - 5 ≥ 0
In this case, the absolute value expression is positive, so we can remove the absolute value sign.
x - 5 ≥ 5
Adding 5 to both sides, we get:
x ≥ 10
Case 2: x - 5 < 0
In this case, the absolute value expression is negative, so we need to reverse the inequality sign and remove the absolute value sign.
-(x - 5) ≥ 5
Expanding the negative sign, we get:
-x + 5 ≥ 5
Subtracting 5 from both sides, we get:
-x ≥ 0
Multiplying both sides by -1 (and reversing the inequality sign since we are multiplying by a negative number), we get:
x ≤ 0
Combining the solutions from both cases, we have:
x ≤ 0 or x ≥ 10
However, if we examine the given answer choices, we see that none of the choices fully match our solution. The correct solution for this inequality is -5 ≤ x ≤ 5.