Answer:
You can proceed as follows:
Step-by-step explanation:
First solve the quadratic inequality [tex]x^{2}-5x+4\leq 0[/tex]. To do that, factorize, then we have that [tex](x-4)(x-1)\leq 0[/tex]. This implies that
[tex]x-1\leq 0\, \text{and}\, x-4\geq 0[/tex]
or
[tex]x-1 \geq 0\, \text{and}\, x-4\leq 0[/tex]
In the first case the solution is the empty set [tex]\emptyset[/tex]. In the second case the solution is the interval [tex]1\leq x \leq 4[/tex]. Now we have that
[tex]A=[1,4][/tex]
[tex]B=(3,5)[/tex]
[tex]C=(3,4][/tex].
To show that [tex]A\cap B\subseteq C[/tex] consider [tex]x\in A\cap B[/tex]. Then [tex]1\leq x \leq 4\, \text{and}\, 1<x<5[/tex], this implies that [tex]3<x\leq 4[/tex], then [tex]x\in C[/tex]. Now, to show that [tex]C\subseteq A\cap B[/tex] consider [tex]x\in C[/tex], then [tex]3<x\leq 4[/tex], then [tex]1\leq x \leq 4\, \text{and}\, 3<x<5[/tex], then [tex]x\in [1,4] \, \text{and}\, x\in (3,5)[/tex], this implies that [tex]x\in A\cap B[/tex].
Observe the image below.
