To fit in an existing frame, the length, $x$, of a piece of glass must be longer than 12 cm but not longer than 12.2 cm. Which inequality can be used to represent the lengths of the glass that will fit in the frame?

A$12 < x \leq 12.2$
B$12 > x \leq 12.2$
C$x > 12$ or $x \leq 12.2$
D$x < 12$ or $x \leq 12.2$

Question

amel65 amel65

02-11-2026
Mathematics

Answered

To fit in an existing frame, the length, $x$, of a piece of glass must be longer than 12 cm but not longer than 12.2 cm. Which inequality can be used to represent the lengths of the glass that will fit in the frame?

A$12 < x \leq 12.2$
B$12 > x \leq 12.2$
C$x > 12$ or $x \leq 12.2$
D$x < 12$ or $x \leq 12.2$

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T15:30:54-05:00

The condition "longer than 12 cm" translates to $x > 12$.
The condition "not longer than 12.2 cm" translates to $x \leq 12.2$.
Combining both requirements results in the compound inequality $12 < x \leq 12.2$.
The correct choice is $\boxed{12 < x \leq 12.2}$.

Explanation

Analysis and Translation of the Problem

The problem asks for an inequality representing the length $x$ of a piece of glass that must be "longer than 12 cm" and "not longer than 12.2 cm".

The phrase "longer than 12 cm" means the length $x$ must be greater than 12, which is written as $x > 12$.
The phrase "not longer than 12.2 cm" means the length $x$ must be less than or equal to 12.2, which is written as $x \leq 12.2$.

For the glass to fit the frame, both conditions must be satisfied simultaneously. This is represented by the intersection of these two inequalities: $x > 12$ and $x \leq 12.2$. Combining these gives the compound inequality $12 < x \leq 12.2$.

Selecting the Correct Inequality

We examine the given options based on our derived inequality $12 < x \leq 12.2$:

The option $12 < x \leq 12.2$ matches our derivation perfectly.
The option $12 > x \leq 12.2$ is mathematically inconsistent.
The option $x > 12$ or $x \leq 12.2$ represents a union of sets, which would include all numbers less than or equal to 12.2, not just the specified range.
The option $x < 12$ or $x \leq 12.2$ also represents a union, covering all numbers less than or equal to 12.2.

Therefore, the correct choice is the first one.

Examples

In engineering and manufacturing, specifying precise tolerances is essential. For instance, a component might be designed to fit into a slot that is slightly larger than the component itself to allow for thermal expansion. If a mechanical part must have a width greater than 10 mm but at most 10.5 mm, this can be expressed as $10 < w \leq 10.5$. This notation clearly defines the acceptable manufacturing range, ensuring the part functions correctly within the assembly without being too loose or too tight.