1 answer:
The absolute value is defined as
So for example, if <em>x</em> = 3, then |<em>x</em>| = |3| = 3, since 3 is positive. On the other hand, if <em>x</em> = -5, then |<em>x</em>| = |-5| = -(-5) = 5, since -5 is negative. The absolute value is always positive.
For the inequality |7 + 8<em>x</em>| > 5, you consider the two cases where the argument to the absolute value (the expression you find inside the bars) is either positive or negative.
• If 7 + 8<em>x</em> ≥ 0, then |7 + 8<em>x</em>| = 7 + 8<em>x</em>, so that
• Otherwise, if 7 + 8<em>x</em> < 0, then |7 + 8<em>x</em>| = -(7 + 8<em>x</em>), so that
The solution to the inequality is the union of these two intervals.
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Answer: The image of the completed table is shown below
In the completed table, we have the following values (left to right)
- Row One: 30, 10, 40
- Row Two: 15, 45, 60
- Row Three: 45, 55, 100
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Explanation:
We're told that "40 people responded that they use mouthwash". That means we'll write "40" at the very end of the "use mouthwash" row. This represents the total number of people who use mouthwash, since we're underneath the "total" in that far right column.
Then we're further told that "Of the people who use mouthwash, 30 people use floss", so we'll write "30" in the first row, first column blank space. These are the people who both floss and use mouthwash. Because earlier we know 40 people use mouthwash overall, this leaves 40-30 = 10 people who use mouthwash, but they don't floss.
To summarize so far: In row 1, we have the following values from left to right: 30, 10, 40
The first two items of this row must add to the last item.
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Now onto row 2
Since 40 people use mouthwash, and 100 people were surveyed, this means that 100-40 = 60 people do not use mouthwash. The value 60 goes at the very end of row 2. This represents another total.
Like earlier, the first two items must add to the last. So 40+60 = 100.
Now turn to the fact that "45 people responded that they don't floss and don't use mouthwash". This tells us that "45" will be in the "no mouthwash" row and "don't floss" column.
We have 60 people who don't use mouthwash, and 45 of those people also don't floss. Therefore, 60-45 = 15 people do floss but they don't use mouthwash. We'll write "15" in the second row, first column.
Along row 2, we have the following values: 15, 45, 60
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Row 3
For each column, add up the first two items to get the third one. This will lead to the totals along this bottom row.
- Floss: 30+15 = 45
- Don't Floss: 10+45 = 55 .... note how this matches up with the third piece of information your teacher provided
The third column is already done, but again we can see that 40+60 = 100.
Furthermore, the first two items of the third row add to 45+55 = 100 which helps confirm our answers.
The full completed table is shown below.
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To summarize, we are effectively just sorting the values given to us to write them into the table. Then we use a bit of algebraic detective work to find out what values are missing so we fill out the completed table.
