All values of x that satisfy ∣x+1∣>x+3 and ∣x+2∣<3 are....
Explanation
The definition of absolute value ∣x+1∣ is
∣x+1∣={x+1,−(x+1),for x+1≥0 or x≥−1for x+1<0 or x<−1
For x≥−1
∣x+1∣=x+1
∣x+1∣>x+3
x+1>x+3
1>3
There is no value of x that satisfies for x≥−1.
For x<−1
∣x+1∣=−(x+1)
∣x+1∣>x+3
−(x+1)>x+3
−x−1>x+3
−2x>4
x<−2
The value of x that satisfies is x<−2. So the solution set is {x<−2}.
Solve the following form ∣x+2∣<3
∣x+2∣<3
−3<x+2<3
−3−2<x<3−2
−5<x<1
So the solution set is {−5<x<1}.
Because the value of x must satisfy the form of ∣x+1∣>x+3 and ∣x+2∣<3, then we make the intersection of both, which is
{x<−2}∩{−5<x<1}={−5<x<−2}