Solve the inequality 2|x - 3| <= 2 + x.
Hence, find x that satisfies 2|x+3|<= 2-x
I solved first one and got the answer as x: 4/3<= x <=8
Then i moved to the second.
By putting x = -x in the first I got,
2|(-x) + 3)| <= 2 + (-x)
===> 2|3 -x | <= 2 - x
===> 2|x-3| <= 2 - x ,
Which gives the range of x;
x:4/3<=x<=8 ,
Is this also the range that of x,
Which satisfies the inequality ,
2|x+3| <= 2-x ???
4/3<=x<=8 is correct
but when you break the absolute value into two choices, you no longer need the || sign.
2|x - 3| <= 2 + x means that
2(x-3) <= 2+x if x-3 >= 0
2(3-x) <= 2+x if x-3 < 0
I use the following method, it never fails.
2|x+3| ≤ 2-x
Initial condition based on definition of ||,
2-x ≥ 0
-x ≥ -2
x ≤ 2
Now for the split,
+2(x+3) ≤ 2-x AND -2(x+3) ≤ 2-x , (had it been ≥, I would have used OR)
3x ≤ -4 AND -x ≤ 8
x ≤ -4/3 and x ≥ -8
-8 ≤ x ≤ -4/3 , which also satisfies the important initial x ≤ 2 condition
To solve the inequality 2|x - 3| <= 2 + x and then find x that satisfies 2|x + 3| <= 2 - x, we can go through the following steps:
1. Start with the inequality 2|x - 3| <= 2 + x.
2. Split the inequality into two cases:
Case 1: (x - 3) >= 0, so |x - 3| = (x - 3).
In this case, the original inequality becomes 2(x - 3) <= 2 + x.
Solving this inequality:
2x - 6 <= 2 + x
x <= 8
Case 2: (x - 3) < 0, so |x - 3| = -(x - 3).
In this case, we need to flip the inequality direction: 2(-x + 3) <= 2 + x.
Solving this inequality:
-2x + 6 <= 2 + x
-3x <= -4
x >= 4/3
3. Combine the results from both cases:
For Case 1: x <= 8
For Case 2: x >= 4/3 (or x >= 1.33)
4. Therefore, the solution to the inequality 2|x - 3| <= 2 + x is: 4/3 <= x <= 8.
Moving on to the second inequality 2|x + 3| <= 2 - x, we can follow a similar process.
1. Start with the inequality 2|x + 3| <= 2 - x.
2. Again, split the inequality into two cases:
Case 1: (x + 3) >= 0, so |x + 3| = (x + 3).
In this case, the original inequality becomes 2(x + 3) <= 2 - x.
Solving this inequality:
2x + 6 <= 2 - x
3x <= -4
x <= -4/3
Case 2: (x + 3) < 0, so |x + 3| = -(x + 3).
In this case, we need to flip the inequality direction: 2(-x - 3) <= 2 - x.
Solving this inequality:
-2x - 6 <= 2 - x
-3x <= 4
x >= -4/3
3. Combine the results from both cases:
For Case 1: x <= -4/3
For Case 2: x >= -4/3
4. Therefore, the solution to the inequality 2|x + 3| <= 2 - x is: x <= -4/3 or x >= -4/3.
Comparing this range with the solution to the first inequality, 4/3 <= x <= 8, we can see that they are different. The range that satisfies the second inequality is x <= -4/3 or x >= -4/3, whereas the range that satisfies the first inequality is 4/3 <= x <= 8.