How many integer values x satisfy the inequality 2<3x−1/6<5?
2<3x−1/6<5
add 1/6 to each part
2+1/6 < 3x < 5+1/6
13/6 < 3x < 31/6
divide by 3
13/18 < x < 31/18
oops, didn't notice the "integer" part till now.
appr .72 < x < appr 1.72
the only integer between .72 and 1.72 is the number 1
So there is one integer that satisfies the inequality
To find the number of integer values that satisfy the inequality 2 < 3x - 1/6 < 5, we need to simplify the inequality and solve it step by step.
Step 1: Remove the fraction by multiplying the entire inequality by 6, the common denominator.
(2)(6) < (3x - 1/6)(6) < (5)(6)
12 < 18x - 1 < 30
Step 2: Add 1 to all parts of the inequality to isolate the variable.
12 + 1 < 18x - 1 + 1 < 30 + 1
13 < 18x < 31
Step 3: Divide all parts of the inequality by 18 to solve for x.
13/18 < (18x)/18 < 31/18
13/18 < x < 31/18
Step 4: Convert the fractions to decimal approximations for a better understanding.
Approximately 0.7222 < x < 1.7222
Step 5: Determine the range of integers within that interval.
The integers that satisfy the inequality are:
x = 1 (only integer value)
Therefore, there is only one integer value, x = 1, that satisfies the given inequality.
To solve the inequality 2 < 3x - 1/6 < 5, we need to find the range of values for x that satisfy the inequality.
Let's solve it step by step:
Step 1: Subtract 1/6 from all parts of the inequality:
2 + 1/6 < 3x - 1/6 + 1/6 < 5 + 1/6
2 + 1/6 < 3x < 5 + 1/6
Step 2: Simplify the expressions:
2 + 1/6 < 3x < 5 + 1/6
13/6 < 3x < 31/6
Step 3: Divide all parts of the inequality by 3:
(13/6) / 3 < x < (31/6) / 3
13/18 < x < 31/18
Now, to find the number of integer values for x that satisfy the inequality, we need to determine the integers that lie within the range of (13/18, 31/18).
To do that, we need to find the smallest integer greater than 13/18 and the largest integer smaller than 31/18.
Finding the smallest integer greater than 13/18:
We can round up the fraction 13/18 to the nearest greater integer, which is 1.
Finding the largest integer smaller than 31/18:
We can round down the fraction 31/18 to the nearest smaller integer, which is 1.
So, the integer values that satisfy the inequality are x = 2, 3, 4, ..., 10.
Therefore, there are 10 - 2 + 1 = <<10-2+1=9>>9 integer values of x that satisfy the given inequality.