Solve the inequality by first rewriting it as an equivalent inequality without absolute value bars. Express solution set using interval notation.
| x+3 | +8 is less than or equal to 14
Thanks so much for helping me!
|x+3| + 8 <= 14,
|x+3| <= 6,
+-(x+3)<= 6,
+(x+3) <= 6,
x+3 >= 6,
X <= 6-3,
X <= 3.
-(x+3) <= 6,
-x - 3 <= 6,
-x <= 6+3,
-x <= 9,
X >= -9.
Solution set: -9 =< X <= 3.
Intrval notation: [-9,3].
To solve the inequality | x+3 | +8 ≤ 14, we can rewrite it as two separate inequalities and solve them individually.
First, let's consider the case when x+3 is nonnegative (greater than or equal to zero):
x+3 + 8 ≤ 14
Simplifying this inequality, we get:
x + 11 ≤ 14
Subtracting 11 from both sides, we have:
x ≤ 3
Now, let's consider the case when x+3 is negative (less than zero):
-(x+3) + 8 ≤ 14
Simplifying this inequality, we get:
- x - 3 + 8 ≤ 14
-x + 5 ≤ 14
Subtracting 5 from both sides:
-x ≤ 9
Multiplying both sides by -1 (remember to flip the inequality sign when multiplying by a negative number):
x ≥ -9
Therefore, the two solutions are x ≤ 3 and x ≥ -9.
To express this in interval notation, we can write it as (-∞, -9] U [3, +∞).
To solve the inequality |x + 3| + 8 ≤ 14, we can start by removing the absolute value bars.
If the expression inside the absolute value bars is positive or zero, we can simply rewrite the inequality without the absolute value as:
x + 3 + 8 ≤ 14
Simplifying:
x + 11 ≤ 14
Next, we can solve for x by isolating it:
x ≤ 14 - 11
x ≤ 3
However, if the expression inside the absolute value bars is negative, we need to change its sign before rewriting the inequality.
So, for x + 3 < 0 (i.e., x < -3), we can rewrite the inequality without the absolute value as:
-(x + 3) + 8 ≤ 14
Simplifying:
- x - 3 + 8 ≤ 14
- x + 5 ≤ 14
Next, solve for x by isolating it:
- x ≤ 14 - 5
- x ≤ 9
To get the final solution, we combine the two solutions:
x ≤ 3 or -x ≤ 9
Since the second inequality is equivalent to x ≥ -9 after multiplying both sides by -1 and flipping the inequality sign, we can rewrite it as:
x ≥ -9
Combining the two inequalities gives us the solution set:
-9 ≤ x ≤ 3
Expressing this solution in interval notation, we write:
[-9, 3]