To find all integer values of [tex]\( x \)[/tex] for which the inequality [tex]\( 12 \leqslant 4x < 25 \)[/tex] holds, follow these steps:
1. First, isolate [tex]\( x \)[/tex] in the inequality.
Start with the given inequality:
[tex]\[
12 \leqslant 4x < 25
\][/tex]
2. Divide each part of the inequality by 4.
Divide the lower bound:
[tex]\[
\frac{12}{4} \leqslant x
\][/tex]
Simplifies to:
[tex]\[
3 \leqslant x
\][/tex]
Divide the upper bound:
[tex]\[
x < \frac{25}{4}
\][/tex]
Simplifies to:
[tex]\[
x < 6.25
\][/tex]
3. Combine the results from both divisions to form a new inequality:
[tex]\[
3 \leqslant x < 6.25
\][/tex]
4. Identify the integer values that satisfy the inequality [tex]\( 3 \leqslant x < 6.25 \)[/tex].
Since [tex]\( x \)[/tex] must be an integer, we look for whole numbers within the given range. These integers are:
[tex]\[
x = 3, 4, 5
\][/tex]
[tex]\( x = 6 \)[/tex] is not included, because [tex]\( 6 \)[/tex] is not less than [tex]\( 6.25 \)[/tex].
In conclusion, the integer values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( 12 \leqslant 4x < 25 \)[/tex] are:
[tex]\[
\boxed{3, 4, 5}
\][/tex]
