Question

2. Find all values of x that satisfies the inequality –
my 2x-3/(x - 2) (x – 4)<0​

Answer 1

We have to find all values of x that satisfies the inequalities (2x - 3)/{(x - 2)(x - 4)} < 0

Solution : step 1 : first equate each term to zero.

i.e., (2x - 3) = 0 ⇒x = 3/2

(x - 2) = 0 ⇒x = 2

(x - 4) = 0 ⇒x = 4

step 2 : putting all gotten values of x (i.e., 3/2, 2, 4) in number line.

Step 3 : See figure , now assume bigger number than 4 ( let's take 6) and put in inequalities,

i.e., (2 × 6 - 3)/(6 - 2)(6 - 4) > 0

so, give positive sign in region 4 to ∞

Step 4 : now take a number between 2 to 4 (see number line),

Let's take 3 between 2 to 4,

We see,. (2 × 3 - 3)/(3 - 2)(3 - 4) < 0

So, give negative sign in region 2 to 4.

Similarly you will get +ve sign in region 3/2 to 2 and -ve sign in region -∞ to 3/2.

Step 6 : now we have to get inequalities of negative sign. So values of x must be the region of negative sign.

i.e., x ∈ (2, 4) U (-∞ , 3/2)