Question

Let A = { x ∈ R : x 2 − 5 x + 4 ≤ 0 } , B = (3 , 5), and C = (3 , 4]. Show that A ∩ B = C . (Recall, you must show two separate statements: A ∩ B ⊆ C and C ⊆ A ∩ B . Hint, you may benefit from factoring the polynomials that appear in the sets.)

Answer 1

Answer:

You can proceed as follows:

Step-by-step explanation:

First solve the quadratic inequality $x^{2}-5x+4\leq 0$. To do that, factorize, then we have that $(x-4)(x-1)\leq 0$. This implies that

$x-1\leq 0\, \text{and}\, x-4\geq 0$

or

$x-1 \geq 0\, \text{and}\, x-4\leq 0$

In the first case the solution is the empty set $\emptyset$. In the second case the solution is the interval $1\leq x \leq 4$. Now we have that

$A=[1,4]$

$B=(3,5)$

$C=(3,4]$.

To show that $A\cap B\subseteq C$ consider $x\in A\cap B$. Then $1\leq x \leq 4\, \text{and}\, 1$, this implies that $3$, then $x\in C$. Now, to show that $C\subseteq A\cap B$ consider $x\in C$, then $3$, then $1\leq x \leq 4\, \text{and}\, 3$, then $x\in [1,4] \, \text{and}\, x\in (3,5)$, this implies that $x\in A\cap B$.

Observe the image below.