To solve this inequality, we need to consider two cases:

When the expression inside the absolute value is non-negative, i.e., when [tex]2x - 7 \geqslant 0[/tex]When the expression inside the absolute value is negative, i.e., when [tex]2x - 7 < 0[/tex]

Case 1: [tex]2x - 7 \geqslant 0[/tex]
Solving for x:
[tex]2x \geqslant 7[/tex]
[tex]x \geqslant \frac{7}{2}[/tex]

Therefore, for this case, the inequality simplifies to:
[tex]2x - 7 \leqslant 5[/tex]
[tex]2x \leqslant 12[/tex]
[tex]x \leqslant 6[/tex]

So, the solution for this case is: [tex]x \geqslant \frac{7}{2}[/tex] and [tex]x \leqslant 6[/tex]

Case 2: [tex]2x - 7 < 0[/tex]
Solving for x:
[tex]2x < 7[/tex]
[tex]x < \frac{7}{2}[/tex]

Therefore, for this case, the inequality simplifies to:
[tex]-(2x - 7) \leqslant 5[/tex]
[tex]-2x + 7 \leqslant 5[/tex]
[tex]-2x \leqslant -2[/tex]
[tex]x \geqslant 1[/tex]

So, the solution for this case is: [tex]x < \frac{7}{2}[/tex] and [tex]x \geqslant 1[/tex]

Combining both cases, the overall solution to the original inequality [tex]|2x - 7| \leqslant 5[/tex] is: [tex]1 \leqslant x \leqslant 6[/tex]