Which inequality represents the compound inequality $x > 3$ and $x < 5$?

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07-11-2026
Mathematics

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Which inequality represents the compound inequality $x > 3$ and $x < 5$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-07T00:08:20-05:00

Interpret the compound inequality $x > 3$ and $x < 5$ as two conditions that must both hold true.
Recognize that $x > 3$ is equivalent to $3 < x$.
Combine the two parts $3 < x$ and $x < 5$ into the single chain $3 < x < 5$.
Conclude that the correct representation is $\boxed{3 < x < 5}$.

Explanation

Understanding the Compound Inequality

The problem asks to represent the compound inequality $x > 3$ and $x < 5$ in a standard compressed notation. A compound inequality joined by "and" requires that the variable $x$ satisfies both given conditions simultaneously. This means $x$ must be strictly greater than $3$ and strictly less than $5$.

Combining the Conditions

We can analyze the two parts of the condition:

The inequality $x > 3$ is equivalent to writing $3 < x$, which states that $3$ is less than $x$.
The inequality $x < 5$ states that $x$ is less than $5$.

By combining these two parts into a single expression, we write the values in ascending order relative to $x$. Since $3 < x$ and $x < 5$, we can merge them into the single chain $3 < x < 5$. This notation compactly expresses that $x$ lies strictly between $3$ and $5$.

Conclusion

The inequality that represents the condition where $x$ is strictly greater than $3$ and strictly less than $5$ is $3 < x < 5$. This matches the given option. Therefore, the equivalent representation is $\boxed{3 < x < 5}$.

Examples

Compound inequalities are frequently used in engineering and manufacturing to specify tolerances for parts. For example, if a machine part is designed to have a length $L$ such that it fits correctly in an assembly, it might need to satisfy $9.95 \text{ cm} < L < 10.05 \text{ cm}$. Using the compressed notation $9.95 < L < 10.05$ ensures that the measurement is within the acceptable range, helping quality control engineers quickly interpret whether a part meets specifications.