Suppose b> d + 1/3 ,c +1<a - 4,
and d + 5/8 > a + 2. Order a, b, c, and
d from greatest to least. Explain your
reasoning.
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Answer:
Let's first write down properties of order used here :
x>y, z>hx>y,z>h implies x+z>y+hx+z>y+h
x>y, y>zx>y,y>z implies x>zx>z
Now let's analyze every inequality we have :
b>d+\frac{1}{3}b>d+
3
1
implies b>db>d , as \frac{1}{3}>0
3
1
>0 and thus b > d+\frac{1}{3} >db>d+
3
1
>d
c+1<a-4c+1<a−4 implies c<ac<a, as c+1<a-4c+1<a−4 implies c<a-5<ac<a−5<a
d+\frac{5}{8}>a+2d+
8
5
>a+2 implies d>ad>a , as d+\frac{5}{8} > a+2d+
8
5
>a+2 implies d>a+1 \frac{3}{8}>ad>a+1
8
3
>a
Thus we have b>d, d>a, a>cb>d,d>a,a>c so we can conclude that b>d>a>cb>d>a>c and the ordering from the greatest to the least is b,d,a,cb,d,a,c .
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