I forgot how to do this problems. Help?
5-|x+4| ≤ -3
≤ is less than or equal to sign
Thanks!
To solve the inequality 5-|x+4| ≤ -3, follow these steps:
Step 1: Isolate the absolute value expression.
To do this, first, add 3 to both sides of the inequality:
5 - |x + 4| + 3 ≤ -3 + 3
This simplifies to:
8 - |x + 4| ≤ 0
Step 2: Remove the absolute value.
To remove the absolute value, we have two cases to consider:
Case 1: x + 4 ≥ 0
In this case, we can simply drop the absolute value symbols:
8 - (x + 4) ≤ 0
Simplify this by distributing the negative sign:
8 - x - 4 ≤ 0
Combine like terms:
4 - x ≤ 0
Case 2: x + 4 < 0
In this case, we need to negate the inequality and the absolute value expression:
8 - (-x - 4) ≤ 0
Simplify this by distributing the negative sign:
8 + x + 4 ≤ 0
Combine like terms:
12 + x ≤ 0
Step 3: Solve for x in each case.
Case 1: 4 - x ≤ 0
Subtract 4 from both sides:
-x ≤ -4
Since we have a negative coefficient of x, we need to reverse the inequality sign:
x ≥ 4
Case 2: 12 + x ≤ 0
Subtract 12 from both sides:
x ≤ -12
Step 4: Combine the solutions from both cases.
Since the inequality is ≤ (less than or equal to), the combined solution will be the intersection of the two individual solutions:
x ≥ 4 and x ≤ -12
Step 5: Final Answer
The final solution to the inequality 5-|x+4| ≤ -3 is:
x ≥ 4 and x ≤ -12