QuestionGiven three positive numbers $x, y, z$ with $x y > z x$ and $y < x$ , determine if $x^{2}$ is greater than, less than, or equal to $z y$ .

Studdy Solution

STEP 1

Assumptions1. $x$ , $y$ , and $z$ are positive numbers. . $x y > z x$ .
3. $y < x$ .

STEP 2

We can rewrite the inequality $x y > z x$ to better understand the relationship between $x$ , $y$ , and $z$ .
$x y - z x >0$

STEP 3

Factor out $x$ from the left side of the inequality.
$x (y - z) >0$

STEP 4

Since $x$ is a positive number, the sign of the inequality will not change when we divide both sides by $x$ .
$y - z >0$

STEP 5

This inequality can be rewritten as $y > z$

STEP 6

Now, we know that $y > z$ and $y < x$ . We can combine these two inequalities to get $z < y < x$

STEP 7

Now, let's compare $x^{2}$ and $z y$ . We know that $x > y > z$ . Since all three numbers are positive, we can say that $x^{2} > x y > z y$ .
So, $x^{2} > z y$ .

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QuestionGiven three positive numbers $x, y, z$ with $x y > z x$ and $y < x$ , determine if $x^{2}$ is greater than, less than, or equal to $z y$ .

Studdy Solution

STEP 1

Assumptions1. $x$ , $y$ , and $z$ are positive numbers. . $x y > z x$ .
3. $y < x$ .

STEP 2

We can rewrite the inequality $x y > z x$ to better understand the relationship between $x$ , $y$ , and $z$ .
$x y - z x >0$

STEP 3

Factor out $x$ from the left side of the inequality.
$x (y - z) >0$

STEP 4

Since $x$ is a positive number, the sign of the inequality will not change when we divide both sides by $x$ .
$y - z >0$

STEP 5

This inequality can be rewritten as $y > z$

STEP 6

Now, we know that $y > z$ and $y < x$ . We can combine these two inequalities to get $z < y < x$

STEP 7

Now, let's compare $x^{2}$ and $z y$ . We know that $x > y > z$ . Since all three numbers are positive, we can say that $x^{2} > x y > z y$ .
So, $x^{2} > z y$ .

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QuestionGiven three positive numbers x,y,zx, y, zx,y,z with xy>zxx y > z xxy>zx and y<xy < xy<x, determine if x2x^{2}x2 is greater than, less than, or equal to zyz yzy.

Studdy Solution

STEP 1

Assumptions1. xxx, yyy, and zzz are positive numbers. . xy>zxx y > z xxy>zx.3. y<xy < xy<x.

STEP 2

We can rewrite the inequality xy>zxx y > z xxy>zx to better understand the relationship between xxx, yyy, and zzz.xy−zx>0x y - z x >0xy−zx>0

STEP 3

Factor out xxx from the left side of the inequality.x(y−z)>0x (y - z) >0x(y−z)>0

STEP 4

Since xxx is a positive number, the sign of the inequality will not change when we divide both sides by xxx.y−z>0y - z >0y−z>0

STEP 5

This inequality can be rewritten asy>zy > zy>z

STEP 6

Now, we know that y>zy > zy>z and y<xy < xy<x. We can combine these two inequalities to getz<y<xz < y < xz<y<x

STEP 7

Now, let's compare x2x^{2}x2 and zyz yzy. We know that x>y>zx > y > zx>y>z. Since all three numbers are positive, we can say that x2>xy>zyx^{2} > x y > z yx2>xy>zy.So, x2>zyx^{2} > z yx2>zy.

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QuestionGiven three positive numbers $x, y, z$ with $x y > z x$ and $y < x$ , determine if $x^{2}$ is greater than, less than, or equal to $z y$ .

Assumptions1. $x$ , $y$ , and $z$ are positive numbers. . $x y > z x$ .
3. $y < x$ .

We can rewrite the inequality $x y > z x$ to better understand the relationship between $x$ , $y$ , and $z$ .
$x y - z x >0$

Factor out $x$ from the left side of the inequality.
$x (y - z) >0$

Since $x$ is a positive number, the sign of the inequality will not change when we divide both sides by $x$ .
$y - z >0$

This inequality can be rewritten as $y > z$

Now, we know that $y > z$ and $y < x$ . We can combine these two inequalities to get $z < y < x$

Now, let's compare $x^{2}$ and $z y$ . We know that $x > y > z$ . Since all three numbers are positive, we can say that $x^{2} > x y > z y$ .
So, $x^{2} > z y$ .