Question

The heights of two buildings are represented by x and y , where x > y .

Use the drop-down menus below to make each comparison true.


x + y ____ 2x

Answer options:

>

<

=


x/y ____ y/x

Answer options:

>

<

=


x+y/y ____ 2

Answer options:

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<

=

Answer 1

`x + y < 2x`

`(x)/(y) > (y)/(x)`

`(x+y)/(y) > 2`

Explanation:

For solving the question first we need to revise the properties of inequality,

a < b ⇒ a ± c < b ± c ∀ a, b, c ∈ R,

a < b ⇒ ca < cb if c > 0,

a < b ⇒ ca > cb if c < 0

Part 1 :

Given,

x and y are two numbers ( must be positive real numbers because they are showing the heights )

Such that, x > y,

Adding x on both sides of the inequality,

x + x > y + x

2x > y + x

⇒ x + y < 2x

Thus, SECOND option is correct.

Part 2 :

∵ x > 0 ⇒ 1/x > 0,

x > y

⇒
`(x)/(x) > (y)/(x)`

`\implies (y)/(x) < 1-----(X)`

Similarly, y > 0 ⇒ 1/y > 0

`(x)/(y)> (y)/(y)`

`\implies (x)/(y) > 1----(Y)`

From equation (X) and (Y),

`(x)/(y) > (y)/(x)`

Thus, FIRST option is correct.

Part 3 :

Now, x > y

x + y > 2y

Since, y > 0 ⇒
`(1)/(y)> 0`

`\implies (x+y)/(y) > 2`

Thus, FIRST option is correct.