|2x+4|<5
|5-3x|<9
What values of x are both of the inequalities above true?
i. -2
ii. -1
iii. 1
I got -1 only and its right because I just plugged it in to find out, but is there another way of solving this perhaps algebraically? For example something like 2x+4<5 and -5? And 5-3x<9 or -9? The you solve for those or something? I tried that but it didn't work...
I get -4/3 < x < 1/2, so only (ii) works.
In this case, it's easy enough just to substitute in each of the values to see which works in both cases.
Always keep in mind the V shape of absolute value graphs. See
http://www.wolframalpha.com/input/?i=plot+y+%3D+|2x%2B4|%2C+y%3D+|5-3x|%2C+y%3D5%2C+y%3D9+for+-2+%3C+x+%3C+2
Yes I got -4/3 with 5-3x<9 and 14/3 with 5-3x>-9. And then same goes for 1/2 with ...<5 and -9/2 with ....>-5. I just don't get why you used -4/3<x<1/2?
For x = - 2
| 2 x + 4 | < 5
| 2 * ( - 2 ) + 4 | < 5
| - 4 + 4 | < 5
| 0 | < 5
0 < 5
Correct
| 5 - 3 x | < 9
| 5 - 3 * ( - 2 ) | < 9
| 5 + 6 | < 9
| 11 | < 9
11 < 9
Not correct
1 correct solution
For x = - 1
| 2 x + 4 | < 5
| 2 * ( - 1 ) + 4 | < 5
| - 2 + 4 | < 5
| 2 | < 5
2 < 5
Correct
| 5 - 3 x | < 9
| 5 - 3 * ( - 1 ) | < 9
| 5 + 3| < 9
| 8 | < 9
8 < 9
Correct
2 correct solutions
For x = 1
| 2 x + 4 | < 5
| 2 * 1 + 4 | < 5
| 2 + 4 | < 5
| 6 | < 5
6 < 5
Not correct
| 5 - 3 x | < 9
| 5 - 3 * 1 | < 9
| 5 - 3 | < 9
2 < 9 Correct
| 5 + 6 | < 9
| 11 | < 9
11 < 9
Not correct
1 correct solution
Answer ii
OK. To solve absolute-value problems, you really have to do them twice. Recall the definition of |z|:
|z| = -z if z < 0
|z| = z if z >= 0
So, you have
|2x+4|<5
That means that
If 2x+4 < 0, (or, x < -2)you have
-(2x+4) < 5
-2x-8 < 5
-2x < 13
x > 13/2
But, this is not a solution, since we started out by assuming x < -2.
Or, if 2x+4 >= 0 (or, x >= -2),
2x+4 < 5
2x < 1
x < 1/2
This is ok, since 1/2 >= -2.
Now you can do the other one in the same way, and wind up with the interval I mentioned at first.
oops - a typo. should be x > -13/2
so, the solution to the first one is -13/2 < x <= 1/2
[10x + 8}v_2
To solve the inequalities algebraically, you can split them into two cases: one for when the expression inside the absolute value is positive, and one for when it is negative.
Let's start with the first inequality: |2x + 4| < 5.
Case 1: (2x + 4) > 0
When the expression inside the absolute value is positive, the inequality becomes:
2x + 4 < 5
Solving this inequality, we get:
2x < 1
x < 1/2
Case 2: (2x + 4) < 0
When the expression inside the absolute value is negative, the inequality becomes:
-(2x + 4) < 5
Solving this inequality, we need to reverse the inequality sign due to multiplying both sides by -1:
2x + 4 > -5
2x > -9
x > -9/2
To combine the two cases, we take the intersection of the solutions, which means we consider the values of x that satisfy both cases. In this case, we look for values of x that are greater than -9/2 but also less than 1/2.
Hence, the solution to the first inequality, |2x + 4| < 5, is -9/2 < x < 1/2.
Now, let's move on to the second inequality: |5 - 3x| < 9.
Case 1: (5 - 3x) > 0
When the expression inside the absolute value is positive, the inequality becomes:
5 - 3x < 9
Solving this inequality, we get:
-3x < 4
x > -4/3
Case 2: (5 - 3x) < 0
When the expression inside the absolute value is negative, the inequality becomes:
-(5 - 3x) < 9
Solving this inequality, we need to reverse the inequality sign due to multiplying both sides by -1:
5 - 3x > -9
-3x > -14
x < 14/3
Again, we take the intersection of the solutions from both cases, which means we consider the values of x that satisfy both cases. In this case, we look for values of x that are greater than -4/3 but also less than 14/3.
Hence, the solution to the second inequality, |5 - 3x| < 9, is -4/3 < x < 14/3.
Now, to find the values of x that satisfy both inequalities, we need to find the intersection of the solution intervals for each inequality.
Let's write the solution intervals side by side:
For |2x + 4| < 5: -9/2 < x < 1/2
For |5 - 3x| < 9: -4/3 < x < 14/3
Now, looking at both intervals, we can see that the only overlap is the interval -4/3 < x < 1/2. Therefore, the values of x that satisfy both inequalities are -4/3 < x < 1/2.
So, the correct answer is ii) -1.
Note: When solving inequalities, it is important to pay attention to the signs when dividing or multiplying both sides by negative numbers. In some cases, the inequality will need to be reversed.