Solve the inequality and express the solution in terms of intervals whenever possible.
| x+3 | < 0.01
I know the answer is : (-3.01,-2.99)
Please show me how to solve this type of math problem.
The absolute value actually represents two cases, the contents are positive (in which case the absolute value does not do much), or if the contents are negative, then the negative sign is dropped.
Mathematically speaking, the above equation can be broken down to two:
x+3 < 0.01 ....(1)
and
-(x+3) < 0.01 ...(2)
Solving (1) gives us
x < 0.01-3 = 2.99, or
x<-2.99
Solving (2) gives us
-(x+3)<0.01
-x-3<0.01
x>-3-0.01=-3.01
So together, we have
x ∈ (-3.01,-2.99)
To solve the inequality |x + 3| < 0.01, we need to consider two cases: when x + 3 is positive and when it is negative. Let's solve each case separately:
Case 1: x + 3 ≥ 0
In this case, |x + 3| simplifies to x + 3. Therefore, the inequality becomes x + 3 < 0.01. To solve for x, subtract 3 from both sides:
x < 0.01 - 3
x < -2.99
Case 2: x + 3 < 0
Now, |x + 3| simplifies to -(x + 3). Therefore, the inequality becomes -(x + 3) < 0.01. Multiply both sides by -1 to change the direction of the inequality:
x + 3 > -0.01
x > -0.01 - 3
x > -3.01
Combining the solutions from both cases, we find that x is within the interval (-3.01, -2.99) to satisfy the inequality |x + 3| < 0.01.