The (absolute value) x-20 (absolute value) is less than or equal to 6. I did it first with six positive and got x is lesss than or equal to 26. Then I made the six negative and got x is less than or equal to 14. But this is impossible for these to both be true, right? How come my book says no solution isn't an answer? (It has 14 is less than or equal to x which is less than or equal to 26 as the answer). How did they get this?
abs(x-20)<=6
but absolute anything is greater than zero, so
0<=abs(x-20)<=6
which now means x can go to 14 to 26, right?
Now, why in the world did you solve it for
abs(x-20)>=-6 If that had be the question, you would be right, however, it is not the problem.
The answer is not a single number, because it is an inequality. Inequalities in the real domain (ℝ) has infinite number of solutions, and is expressed in the form of an interval.
When dealing with the absolute value function, it makes life easier to split the inequality into two equations. After that, the two solution sets will be intersected to give the final solution.
For example, to solve
|x-3| ≤ 2
we write
x-3 ≤ 2 ....(1), and
-(x-3) ≤2 ....(2).
The solution of (1) gives
x≤5
and the solution of (2) gives
x≥1
So the answer would be
1≤x≤5, or expressin interval notation,
x ∈ [1,5]
To solve the inequality |x - 20| ≤ 6, you correctly considered two cases: one where x - 20 is positive and another where it is negative.
In the first case, you correctly reasoned that if x - 20 is positive, then the inequality becomes x - 20 ≤ 6. By adding 20 to both sides of the inequality, you can simplify it to x ≤ 26. This means that any value of x that is less than or equal to 26 satisfies the inequality.
In the second case, when x - 20 is negative, you wrote the inequality as -(x - 20) ≤ 6, which can be simplified to -x + 20 ≤ 6. By subtracting 20 from both sides, you get -x ≤ -14. However, you made a sign error; the correct answer should be x ≥ 14 (not x ≤ 14). This tells us that any value of x that is greater than or equal to 14 satisfies the inequality.
Now, let's consider the combined result from both cases. We found that x could be less than or equal to 26 (from x - 20 ≤ 6) or greater than or equal to 14 (from -x + 20 ≤ 6).
The solution to the inequality is the intersection of these two cases, which means the values of x that satisfy BOTH conditions. When you plot this on a number line, you'll find that the overlapping region is from 14 to 26, inclusive. Therefore, the answer to the inequality is 14 ≤ x ≤ 26.
It seems like there was a mistake in the book answer where they only wrote 14 ≤ x ≤ 26. The correct solution is to have the "less than or equal to" symbol on both ends.
To solve the inequality |x - 20| ≤ 6, you need to consider two cases: when x - 20 is positive and when x - 20 is negative.
1. When x - 20 is positive:
In this case, the inequality becomes x - 20 ≤ 6. To solve for x, you can add 20 to both sides of the inequality, giving x ≤ 26.
2. When x - 20 is negative:
In this case, the inequality becomes -(x - 20) ≤ 6. To solve for x, you need to distribute the negative sign, giving -x + 20 ≤ 6. Then, subtracting 20 from both sides gives -x ≤ -14. Multiplying both sides by -1 (while flipping the inequality symbol) gives x ≥ 14.
Based on the solutions from both cases, you found that x ≤ 26 and x ≥ 14. Therefore, the intersection of these two solution sets is 14 ≤ x ≤ 26.
Regarding your question about why the book says there is no solution, it seems to be incorrect. From the calculations, it is clear that there is indeed a solution: 14 ≤ x ≤ 26. You have correctly determined that it is impossible for both statements "x ≤ 26" and "x ≥ 14" to be true. However, it is possible for both statements to be true when combined with the logical operator "and" (denoted by the symbol ≤).