Question

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? 12 < x ≤ 12.2; 12 > x ≤ 12.2; x > 12 or x ≤ 12.2; x < 12 or x ≤ 12.2

Explanation:

Step1: Analyze "longer than 12 cm"

"Longer than 12 cm" means \( x > 12 \) (since \( x \) has to be more than 12, not equal to 12).

Step2: Analyze "not longer than 12.2 cm"

"Not longer than 12.2 cm" means \( x \leq 12.2 \) (since \( x \) can be equal to 12.2 or less than 12.2).

Step3: Combine the inequalities

We need to find the inequality that represents both conditions at the same time (since the length has to satisfy both "longer than 12" and "not longer than 12.2"). So we combine \( x > 12 \) and \( x \leq 12.2 \) into \( 12 < x \leq 12.2 \).

Now let's check the other options:

• Option \( 12 > x \leq 12.2 \): This would mean \( x \) is less than 12 and at most 12.2, which contradicts "longer than 12".
• Option \( x > 12 \) or \( x \leq 12.2 \): "Or" means either condition, but we need both conditions to be true (the length has to be both longer than 12 AND not longer than 12.2, not one or the other).
• Option \( x < 12 \) or \( x \leq 12.2 \): This includes lengths less than 12, which contradicts "longer than 12".

Answer:

\( 12 < x \leq 12.2 \) (the first option: \( 12 < x \leq 12.2 \))