Solve each inequality.
1). |z -6| + 8 > 5
A: No solution?
2). 2|x -3| <= 4
A: 1 <= x <= 5
|z-6|> -3
No solution
x -3 <= 2
x <= 5
1<=x <=5
|z-6|> -3
No solution
x -3 <= 2
x <= 5
1<=x <=5
actually
|z-6| > -3 for ALL values of z
try putting in some values of z, you will see that everything works.
What do you mean?
What do you mean by "What do you mean" ??
did you try some values of z like I suggested ?
e.g. z = 45
|45 - 4| > -3 ??? sure is, so z = 45 works
z = 0
| 0-4| > -3
4 > -3 ??? sure is, so z = 0
z = -1000
|-1000 - 4| > -3
1004 > -3 , sure is, so z = -1000
etc ....
So it is true for any or all values of z you pick
If you can find a value of z which makes the statement false, I will send you $5.00
To solve inequalities involving absolute values, we need to break them down into two separate cases: when the expression inside the absolute value is positive and when it is negative.
Let's go through each inequality step by step:
1) |z - 6| + 8 > 5
Subtracting 8 from both sides, we get:
|z - 6| > -3
Now, since an absolute value is always non-negative, the expression inside the absolute value must be greater than -3.
Case 1: z - 6 > -3
Adding 6 to both sides, we have:
z > 3
Case 2: -(z - 6) > -3
Multiplying through by -1 (which flips the inequality sign), we get:
z - 6 < 3
Adding 6 to both sides, we have:
z < 9
Therefore, the solution for this inequality is z > 3 or z < 9. Since these two conditions cannot be simultaneously true, there is no solution for this inequality.
2) 2|x - 3| ≤ 4
Dividing both sides by 2, we have:
|x - 3| ≤ 2
Now, we need to consider the two cases.
Case 1: x - 3 ≥ 0 (when the expression inside the absolute value is positive)
In this case, we have:
x - 3 ≤ 2
Adding 3 to both sides, we get:
x ≤ 5
Case 2: x - 3 < 0 (when the expression inside the absolute value is negative)
In this case, we need to use the fact that multiplying an inequality by -1 flips the sign. So we have:
- (x - 3) ≤ 2
Expanding and simplifying, we get:
- x + 3 ≤ 2
Subtracting 3 from both sides, we have:
- x ≤ -1
Dividing through by -1 (which also flips the sign), we get:
x ≥ 1
Combining the solutions from both cases, we have:
1 ≤ x ≤ 5
Therefore, the solution for this inequality is 1 ≤ x ≤ 5.