Mathematics: Can ya’ll help me out

Can ya’ll help me out - id13912760, gabby7398

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21.03.2020

x + y < 2x

$\frac{x}{y} \frac{y}{x}$

$\frac{x+y}{y} 2$

Step-by-step 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

⇒ $\frac{x}{x} \frac{y}{x}$

$\implies \frac{y}{x} < 1-----(X)$

Similarly, y > 0 ⇒ 1/y > 0

$\frac{x}{y} \frac{y}{y}$

$\implies \frac{x}{y} 1----(Y)$

From equation (X) and (Y),

$\frac{x}{y} \frac{y}{x}$

Thus, FIRST option is correct.

Part 3 :

Now, x > y

x + y > 2y

Since, y > 0 ⇒ $\frac{1}{y} 0$

$\implies \frac{x+y}{y} 2$

Thus, FIRST option is correct.

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