2 quadratic inequation questions

Question

i dont understand this topic at all could you give me a rule to follow for these q's

 

n^2 - b > or equal to 0

 

3x^2 < or equal to 7x - 4

Answer 1

n² - b ≥0  ≤

n2 ≥b, assuming b ≥0.

|n|≥b 

n ≤-b and/or n≥b

 3x2≤7x-2

3x²-7x+2≤0 (factor LHS)

(x-2)(3x-1)≤0 

Now, product of two factor is ≤0 is they have opposite signs:

a)  x-2≥0     ---> x≥2

     3x-1≤0   ---> x≤1/3  

Both inequalities here cannot be satisfied simultaneously. So x belongs to empty set.

 

b)  x-2≤0    -->x≤2

     3x-1≥0  ---> x≥1/3

From these it follows that  1/3 ≤ x≤2.