Answer:
<em>Option C</em>
Step-by-step explanation:
Consider each of these graphs. Let us formulate an inequality for each of them, and match the one with an inequality of
;
![Graph 1, x > 13.5\\Graph 2, x \geq 14\\Graph 3, x > 14.5\\Graph 4, x \geq 15](https://tex.z-dn.net/?f=Graph%201%2C%20x%20%3E%2013.5%5C%5CGraph%202%2C%20x%20%5Cgeq%2014%5C%5CGraph%203%2C%20x%20%3E%2014.5%5C%5CGraph%204%2C%20x%20%5Cgeq%2015)
Graph 3 is the only one that matches with the inequality provided to us.
* Note that shaded circles are represented by a greater / less than or equal to, and non - shaded circles are represented by a greater / less than sign.
<em>Solution ⇒ Graph 3</em>
The first answer (question 4) is 12 and the second answer (question 5) is 13 hope this helps!!!
Answer:
a. 87.5%
b. 30 g of 12 carat and 60 g of 21 carat.
Step-by-step explanation:
a. The percentage of gold in 21 carat is given by:
![gold_{21} = \frac{21}{24}*100 = 87.5 \%](https://tex.z-dn.net/?f=gold_%7B21%7D%20%3D%20%5Cfrac%7B21%7D%7B24%7D%2A100%20%3D%2087.5%20%5C%25)
b. Since the mass he wants to produce is 90 g, then the added mass of the 12 carat, "x", and the mass of the 21 carat "y", must be equal to that:
![x + y = 90](https://tex.z-dn.net/?f=x%20%2B%20y%20%3D%2090)
The sum of gold on each carat type must be equal to the amount of gold on the final mixture, since a 12 carat has 50% gold and a 21 carat has 87.5% gold, then we have:
![0.5*x + 0.875*y = 0.75*90\\\\0.5*x + 0.875*y = 67.5\\](https://tex.z-dn.net/?f=0.5%2Ax%20%2B%200.875%2Ay%20%3D%200.75%2A90%5C%5C%5C%5C0.5%2Ax%20%2B%200.875%2Ay%20%3D%2067.5%5C%5C)
We now have a system of equations which we can solve to find the necessary mass of each type.
![x + y = 90 \text{ } *-0.5\\0.5*x + 0.875*y = 67.5\\\\-0.5*x - 0.5*y = -45\\0.5*x + 0.875*y = 67.5\\\\0.375*y = 22.5\\\\y = 60](https://tex.z-dn.net/?f=x%20%2B%20y%20%3D%2090%20%5Ctext%7B%20%7D%20%2A-0.5%5C%5C0.5%2Ax%20%2B%200.875%2Ay%20%3D%2067.5%5C%5C%5C%5C-0.5%2Ax%20-%200.5%2Ay%20%3D%20-45%5C%5C0.5%2Ax%20%2B%200.875%2Ay%20%3D%2067.5%5C%5C%5C%5C0.375%2Ay%20%3D%2022.5%5C%5C%5C%5Cy%20%3D%2060)
![x = 90 - y = 90 - 60\\\\x = 30](https://tex.z-dn.net/?f=x%20%3D%2090%20-%20y%20%3D%2090%20-%2060%5C%5C%5C%5Cx%20%3D%2030)
He needs 30 g of 12 carat gold and 60 g of 21 carat gold.
Answer:
28 and 29
Step-by-step explanation:
28x29=812
28+29=57
The result follows directly from properties of modular arithmetic:
![b\equiv5\pmod{12}\implies 7b\equiv35\equiv-1\equiv\boxed{11}\pmod{12}](https://tex.z-dn.net/?f=b%5Cequiv5%5Cpmod%7B12%7D%5Cimplies%207b%5Cequiv35%5Cequiv-1%5Cequiv%5Cboxed%7B11%7D%5Cpmod%7B12%7D)
That is,
![b\equiv5\pmod{12}](https://tex.z-dn.net/?f=b%5Cequiv5%5Cpmod%7B12%7D)
means we can write
for some integer
. Then
![7b=7(12n+5)=12(7n)+35](https://tex.z-dn.net/?f=7b%3D7%2812n%2B5%29%3D12%287n%29%2B35)
and taken mod 12, the first term goes away, so
![7b\equiv35\pmod{12}](https://tex.z-dn.net/?f=7b%5Cequiv35%5Cpmod%7B12%7D)
etc