How many integer values x satisfy the inequality 2<3x−1/6<5 ?
Wrong answer
To solve the inequality 2 < 3x - 1/6 < 5, we will break it down into two separate inequalities:
1) 2 < 3x - 1/6
2) 3x - 1/6 < 5
Let's solve each inequality separately:
1) 2 < 3x - 1/6
First, we add 1/6 to both sides:
2 + 1/6 < 3x
Simplifying,
12/6 + 1/6 < 3x
13/6 < 3x
Dividing both sides by 3,
13/18 < x
So, the first inequality can be written as: 13/18 < x.
2) 3x - 1/6 < 5
First, we add 1/6 to both sides:
3x < 5 + 1/6
Simplifying,
3x < 30/6 + 1/6
3x < 31/6
Dividing both sides by 3,
x < 31/18
So, the second inequality can be written as: x < 31/18.
Now, combining both inequalities, we have:
13/18 < x < 31/18
To find the number of integer values of x that satisfy this inequality, we need to find how many integers fall within this range.
To determine this, we can subtract the floor of the lower limit from the ceiling of the upper limit:
Number of integer values of x = ⌈31/18⌉ - ⌊13/18⌋
Simplifying,
Number of integer values of x = 2 - 0
Therefore, there are 2 integer values of x that satisfy the inequality.
To solve the inequality 2 < 3x - 1/6 < 5, we can follow these steps:
Step 1: Solve the first inequality, 2 < 3x - 1/6.
Add 1/6 to both sides:
2 + 1/6 < 3x
Multiply by the reciprocal of 3 to isolate x:
(2 + 1/6) * (1/3) < x
Simplify the left side:
13/6 * 1/3 < x
13/18 < x
Step 2: Solve the second inequality, 3x - 1/6 < 5.
Add 1/6 to both sides:
3x < 5 + 1/6
Multiply by the reciprocal of 3 to isolate x:
(5 + 1/6) * (1/3) > x
Simplify the left side:
31/6 * 1/3 > x
31/18 > x
Therefore, we have found that the value of x satisfies the inequality 13/18 < x < 31/18.
Now, to find the number of integer values x that satisfy the inequality, we need to identify the number of integers within this range.
Step 3: Find the number of integers between 13/18 and 31/18.
First, find the smallest integer greater than or equal to 13/18. This can be done by rounding up 13/18 to the next integer.
ceil(13/18) = 1, so the smallest integer greater than or equal to 13/18 is 1.
Next, find the largest integer less than or equal to 31/18. This can be done by rounding down 31/18 to the previous integer.
floor(31/18) = 1, so the largest integer less than or equal to 31/18 is 1.
Since the range is from 1 to 1, there is only one integer value x that satisfies the inequality.
Therefore, there is one integer value x that satisfies the inequality 2 < 3x - 1/6 < 5.
2<3x−1/6<5
add 1/6
2+1/6 < 3x < 5+1/6
13/6 < 3x < 31/6
divide by 3
13/18 < x < 31/18
.7222 < x < 1.722
the only integer between .722 and 1.722 is the integer 1
So there is only one integer satisfying the inequality.