Find an integer $x$ such that $\frac{2}{3} < \frac{x}{5} < \frac{6}{7}$
2/3 < x/5 < 6/7
10/3 < x < 30/7
3.3 < x < 4.3
Looks like x=4
Well, according to my calculations, the only integer that satisfies the given inequality is $x = 3.33...$
Just kidding! Since $x$ has to be an integer, we can't have decimals involved. Let's examine each fraction separately:
For $\frac{2}{3} < \frac{x}{5}$, if we multiply both sides by $5$, we get $\frac{10}{3} < x$. So, $x$ must be greater than $\frac{10}{3}$.
For $\frac{x}{5} < \frac{6}{7}$, if we multiply both sides by $5$, we get $x < \frac{30}{7}$. So, $x$ must be less than $\frac{30}{7}$.
Combining the two inequalities, we have $\frac{10}{3} < x < \frac{30}{7}$. To find an integer $x$ within this range, we can use the nearest integer function. Thus, $x$ can be either $4, 5, 6, 7, 8,$ or $9$.
So, the answer is that any of these integers $x = 4, 5, 6, 7, 8,$ or $9$ will satisfy the given inequality.
To find an integer $x$ that satisfies $\frac{2}{3} < \frac{x}{5} < \frac{6}{7}$, we can perform the following steps:
Step 1: Multiply all terms of the inequality by 5 to eliminate the denominator on the left-hand side and the right-hand side:
\[
\frac{2}{3} \cdot 5 < \frac{x}{5} \cdot 5 < \frac{6}{7} \cdot 5 \implies \frac{10}{3} < x < \frac{30}{7}
\]
Step 2: Convert the fractions to mixed numbers to get a better understanding of the values:
\[
\frac{10}{3} = 3\frac{1}{3} \quad \text{and} \quad \frac{30}{7} = 4\frac{2}{7}
\]
Step 3: Since we are looking for an integer $x$, we need to find the integer values within the range $3\frac{1}{3} < x < 4\frac{2}{7}$.
Step 4: The possible integer values for $x$ in this range are 4 and 5. Therefore, the solution is $x = 4$ or $x = 5$.
To find an integer $x$ such that $\frac{2}{3} < \frac{x}{5} < \frac{6}{7}$, we need to analyze the given inequality and determine the possible integer values for $x$.
Let's start by focusing on the left side of the inequality, $\frac{2}{3} < \frac{x}{5}$. To eliminate the fractions and make the comparison easier, we can multiply both sides of the inequality by the common denominator of 3 and 5, which is 15:
$15 \cdot \frac{2}{3} < 15 \cdot \frac{x}{5}$
Simplifying this, we obtain:
$10 < 3x$
Next, we need to solve the inequality $10 < 3x$. We can divide both sides of the inequality by 3 to isolate the value of $x$:
$\frac{10}{3} < x$
Since we are looking for an integer value for $x$, we can round the right side up to the next integer:
$x > \frac{10}{3} \approx 3.\overline{3}$
Therefore, $x$ must be greater than or equal to 4.
Now, let's focus on the right side of the inequality, $\frac{x}{5} < \frac{6}{7}$. Similarly, we can eliminate the fractions by multiplying both sides by the common denominator of 5 and 7, which is 35:
$35 \cdot \frac{x}{5} < 35 \cdot \frac{6}{7}$
Simplifying this, we get:
$7x < 30$
Next, solve the inequality $7x < 30$ by dividing both sides by 7:
$x < \frac{30}{7}$
Since we are looking for an integer value for $x$, we can round the right side down to the previous integer:
$x < \frac{30}{7} \approx 4.2857$
Therefore, $x$ must be less than 4.
Combining the results from the two inequalities, we find that $x$ must be an integer greater than or equal to 4 and less than 4. Since there is no integer that satisfies both conditions, there is no solution for the given inequality.
Hence, there is no integer $x$ that satisfies $\frac{2}{3} < \frac{x}{5} < \frac{6}{7}$.