Question

Given
$P=\left\{x:5<2x-1\le 11,xϵR\right\}$, $Q=\left\{x:-1\le 3+4x<23,xϵZ\right\}$. Find the number of integral solution of $P\cap Q$.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is A 1

Given: $P=5<2x-1\le 11$

$P=5<2x-1\le 11$

Let's separate the given inequation in two inequation.

Rule: If a term of an inequation is transferred from one side to the other side of the inequation, the sign of the term gets changed. Let's apply this rule in the above inequation.

$5<2x-1$ $2x-1\le 11$
$6<2x$ $2x\le 12$
$3<x$ $x\le 6$

$\therefore 3<x\le 6$

Solution Set for P = {4,5,6}

$Q=-1\le 3+4x<23$

Let's separate the given inequation in two inequation.

Rule: If a term of an inequation is transferred from one side to the other side of the inequation, the sign of the term gets changed. Let's apply this rule in the above inequation.

$-1\le 3+4x$ $3+4x<23$
$-4\le 3+4x$ $4x<20$
$-1\le x$ $x<5$

$-1\le x<5$

Q= { -1, 0, 1, 2, 3, 4}

P$\cap Q$ = {4}

So, $P\cap Q$ = 4

Hence there is only one integral solution and that is 4.