Đáp án:

$a)M = 3x - 4$

$b)N = 2x + 5$

$c)P = \left\{ \begin{array}{l}
4x - 7,x \ge 2\\
2x - 3,\dfrac{5}{3} \le x < 2\\
 - 4x + 7,x < \dfrac{5}{3}
\end{array} \right.$

$d)Q = \left\{ \begin{array}{l}
 - 9x - 3,x \ge 3\\
3 - 11x,0 \le x < 3\\
9x + 3,x < 0
\end{array} \right.$

Giải thích các bước giải:

$\begin{array}{l}
a)x > 1,5 \Leftrightarrow x > \dfrac{3}{2} \Rightarrow \left\{ \begin{array}{l}
2x - 3 > 0\\
x - 1 > 0
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
\left| {2x - 3} \right| = 2x - 3\\
\left| {x - 1} \right| = x - 1
\end{array} \right.\\
 \Rightarrow M = \left| {2x - 3} \right| + \left| {x - 1} \right| = 2x - 3 + x - 1 = 3x - 4\\
b)x <  - 1 \Rightarrow \left\{ \begin{array}{l}
x + 1 < 0\\
2 - x > 3 > 0
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
\left| {x + 1} \right| =  - x - 1\\
\left| {2 - x} \right| = 2 - x
\end{array} \right.\\
 \Rightarrow N = \left| {2 - x} \right| - 3\left| {x + 1} \right| = 2 - x - 3\left( { - x - 1} \right) = 2x + 5\\
c)P = \left| {3x - 5} \right| + \left| {x - 2} \right|\\
 + )x \ge 2\\
 \Rightarrow \left\{ \begin{array}{l}
\left| {3x - 5} \right| = 3x - 5\\
\left| {x - 2} \right| = x - 2
\end{array} \right.\\
 \Rightarrow P = \left| {3x - 5} \right| + \left| {x - 2} \right| = 3x - 5 + x - 2 = 4x - 7\\
 + )\dfrac{5}{3} \le x < 2\\
 \Rightarrow \left\{ \begin{array}{l}
\left| {3x - 5} \right| = 3x - 5\\
\left| {x - 2} \right| = 2 - x
\end{array} \right.\\
 \Rightarrow P = \left| {3x - 5} \right| + \left| {x - 2} \right| = 3x - 5 + 2 - x = 2x - 3\\
 + )x < \dfrac{5}{3}\\
 \Rightarrow \left\{ \begin{array}{l}
\left| {3x - 5} \right| =  - 3x + 5\\
\left| {x - 2} \right| = 2 - x
\end{array} \right.\\
 \Rightarrow P = \left| {3x - 5} \right| + \left| {x - 2} \right| =  - 3x + 5 + 2 - x =  - 4x + 7\\
d)Q = \left| {x - 3} \right| - 2\left| { - 5x} \right| = \left| {x - 3} \right| - 10\left| x \right|\\
 + )x \ge 3\\
 \Rightarrow \left\{ \begin{array}{l}
\left| {x - 3} \right| = x - 3\\
\left| x \right| = x
\end{array} \right.\\
 \Rightarrow Q = \left| {x - 3} \right| - 10\left| x \right| = x - 3 - 10x =  - 9x - 3\\
 + )0 \le x < 3\\
 \Rightarrow \left\{ \begin{array}{l}
\left| {x - 3} \right| = 3 - x\\
\left| x \right| = x
\end{array} \right.\\
 \Rightarrow Q = \left| {x - 3} \right| - 10\left| x \right| = 3 - x - 10x = 3 - 11x\\
 + )x < 0\\
 \Rightarrow \left\{ \begin{array}{l}
\left| {x - 3} \right| = 3 - x\\
\left| x \right| =  - x
\end{array} \right.\\
 \Rightarrow Q = \left| {x - 3} \right| - 10\left| x \right| = 3 - x + 10x = 9x + 3
\end{array}$