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How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?
- a)3
- b)4
- c)5
- d)6
- e)7
Correct answer is option 'C'. Can you explain this answer?
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How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?a...
To solve the inequality |x - 5| ≤ 2.5, we can consider two cases:
Case 1: x - 5 ≥ 0
In this case, the absolute value |x - 5| is equal to x - 5.
Substituting this into the inequality, we have x - 5 ≤ 2.5.
By adding 5 to both sides of the inequality, we get x ≤ 7.5.
In this case, the absolute value |x - 5| is equal to x - 5.
Substituting this into the inequality, we have x - 5 ≤ 2.5.
By adding 5 to both sides of the inequality, we get x ≤ 7.5.
Case 2: x - 5 < 0
In this case, the absolute value |x - 5| is equal to -(x - 5) = -x + 5.
Substituting this into the inequality, we have -x + 5 ≤ 2.5.
By subtracting 5 from both sides of the inequality and multiplying by -1, we get x ≥ 2.5.
In this case, the absolute value |x - 5| is equal to -(x - 5) = -x + 5.
Substituting this into the inequality, we have -x + 5 ≤ 2.5.
By subtracting 5 from both sides of the inequality and multiplying by -1, we get x ≥ 2.5.
Combining the results from both cases, we have 2.5 ≤ x ≤ 7.5.
To find the number of integer values of x that satisfy the inequality, we can count the integers within this range.
The integers that satisfy the inequality are 3, 4, 5, 6, and 7. (Note that 2.5 and 7.5 are not included since the inequality is not inclusive.)
Therefore, there are 5 integer values of x that satisfy the inequality.
Thus, the correct answer is C: 5.
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How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?a...
Understanding the Inequality:
The inequality given is |x - 5| ≤ 2.5. This means that the distance between x and 5 should be less than or equal to 2.5.
Finding the Possible Values of x:
To solve this inequality, we can break it down into two separate inequalities:
1. x - 5 ≤ 2.5
2. x - 5 ≥ -2.5
Solving the First Inequality:
x - 5 ≤ 2.5
Adding 5 to both sides, we get:
x ≤ 7.5
Solving the Second Inequality:
x - 5 ≥ -2.5
Adding 5 to both sides, we get:
x ≥ 2.5
Finding the Intersection of Solutions:
The values of x should satisfy both x ≤ 7.5 and x ≥ 2.5.
Therefore, the possible values of x lie within the range 2.5 ≤ x ≤ 7.5.
Calculating the Number of Integer Values:
The integer values that satisfy this range are 3, 4, 5, 6, and 7.
Hence, there are 5 integer values of x that satisfy the given inequality.
Therefore, the correct answer is option C. 5.
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