Please help on two math questions. I don’t understand how to do them.
1 answer:
When you represent intervals on the number line, you're including full dots, excluding empty dots, and you're considering numbers highlighted by the line.
In the first case, you've highlighted everything before -2 (full dot, thus included), and everything after 1 (empty dot, excluded). So, the set would be
or, in interval notation,
In the second case, you are looking for all numbers between -3 and 5. This interval is symmetric with respect to 1: you're considering all numbers that are at most 4 units away from 1, both to the left and to the right.
This means that the difference between your numbers at 1 must be at most 4, which is modelled by
where the absolute values guarantees that you'll pick numbers to the left and to the right of 1.
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Let's call our unknown quantity 
. We'll need to add lbs of chocolate with a 1/15 ratio of cocoa butter to caramel to 300 lbs of chocolate with a 1/7 ratio of cocoa butter to caramel to yield 
lbs of chocolate with a 1/9 ratio of cocoa butter to caramel. We can set this up as an equation, but it's important to note something first: the ratio of the <em>caramel to the cocoa butter</em> is provided, but this is not the same thing as the <em>fraction of the chocolate the cocoa butter takes up</em>.
The 300 pounds of chocolate have 1 + 7 = 8 parts of cocoa butter and caramel total, which means that cocoa butter takes up 1/8 of those total parts; the pounds we're adding on has 1 + 15 = 16 total parts, which means the cocoa butter takes up 1/16 of those; and the pounds being produced have 1 + 9 = 10 total parts, so the cocoa butter takes up 1/10 of those parts. With this in mind, we can set up the following equation:
which we can rewrite as
From here, it would be helpful to combine the fractions on the left side of the equation. To do this, we'll convert the denominator of
to 16, multiplying it by 
to obtain the fraction 
. Combining that with 
, we have:
To get rid of the denominator on the left, we'll multiply both sides of the equation by 16, and to eliminate the one on the right, we'll multiply both sides by 10:
Simplifying:
And finally, we solve for:
So, the confectioner needs 200 lbs of chocolate that's 1 part cocoa butter and 15 parts caramel.
The 300 pounds of chocolate have 1 + 7 = 8 parts of cocoa butter and caramel total, which means that cocoa butter takes up 1/8 of those total parts; the
which we can rewrite as
From here, it would be helpful to combine the fractions on the left side of the equation. To do this, we'll convert the denominator of
To get rid of the denominator on the left, we'll multiply both sides of the equation by 16, and to eliminate the one on the right, we'll multiply both sides by 10:
Simplifying:
And finally, we solve for
So, the confectioner needs 200 lbs of chocolate that's 1 part cocoa butter and 15 parts caramel.