Question

Qx Find a value of \( K>0 \) such that the implication \[ |x+4| \leq 2 \Rightarrow\left|\frac{(x+1)^{2}}{x+7}-3\right|>K|x+4| \text {. } \] is true. Make sure you justify your choice of \( K \), with

Answer 1

The value of
`\(K\)` that satisfies the given implication for all
`\(x\)` is
`\(K = (1)/(2)\)`.

To find a value of
`\(K\)` such that the implication

`\[|x+4| \leq 2 \Rightarrow \left|((x+1)^2)/(x+7)-3\right| > K|x+4|\]`

is true, we need to analyze the given inequality for different cases of
`\(x+4\)`.

Step 1: Analyzing
`\(x+4\)`:

Case 1:
`\(x+4 > 0\)`

In this case,
`\(|x+4| = x+4\)`.

Case 2:
`\(x+4 < 0\)`

In this case,
`\(|x+4| = -(x+4) = -x-4\)`.

Step 2: Analyzing
`\(|x+4| \leq 2\)`:

For Case 1:
`\(x+4 > 0\)`

We have
`\(|x+4| \leq 2 \Rightarrow x+4 \leq 2 \Rightarrow x \leq -2\)`.

For Case 2:
`\(x+4 < 0\)`

We have
`\(|x+4| \leq 2 \Rightarrow -x-4 \leq 2 \Rightarrow -x \leq 6 \Rightarrow x \geq -6\)`.

So, the condition
`\(|x+4| \leq 2\)` is satisfied when
`\(x \leq -2\)` or
`\(x \geq -6\)`.

Step 3: Analyzing the inequality for each case:

Case 1:
`\(x \leq -2\)`

In this case,
`\(|x+4| = x+4\)` and the given inequality becomes:

`\[x+4 \leq 2 \Rightarrow x \leq -2\]`

Now, let's analyze the right side of the inequality:

`\[\left|((x+1)^2)/(x+7)-3\right| > K|x+4|\]`

For
`\(x \leq -2\)`, we know that
`\(x+7\)` is also negative. So, let's consider two subcases:

Subcase 1:
`\(x+7 < 0\)`

In this subcase,
`\(((x+1)^2)/(x+7)-3\)` is negative, and the inequality becomes:

`\[-\left(((x+1)^2)/(x+7)-3\right) > K(x+4)\]`

Now, simplify the left side:

`\[3 - ((x+1)^2)/(x+7) > K(x+4)\]`

Subcase 2:
`\(x+7 > 0\)`

In this subcase,
`\(((x+1)^2)/(x+7)-3\)` is positive, and the inequality becomes:

`\[((x+1)^2)/(x+7)-3 > K(x+4)\]`

Now, let's analyze each subcase separately.

Subcase 1:

`\[3 - ((x+1)^2)/(x+7) > K(x+4)\]`

To find the maximum value of
`\(x+4\)` for
`\(x \leq -2\)`, we need to maximize
`\(x\)` by setting
`\(x = -2\)`. Thus, the maximum value of
`\(x+4\)` is
`\(2\)`.

So, the inequality becomes:

`\[3 - ((x+1)^2)/(x+7) > 2K\]`

Subcase 2:

`\[((x+1)^2)/(x+7)-3 > K(x+4)\]`

To find the minimum value of
`\(x+4\)` for
`\(x \leq -2\)`, we need to minimize
`\(x\)` by setting
`\(x = -2\)`. Thus, the minimum value of
`\(x+4\)` is
`\(0\)`.

So, the inequality becomes:

`\[((x+1)^2)/(x+7)-3 > 0\]`

Now, we have analyzed the inequality for both subcases:

Subcase 1:
`\(3 - ((x+1)^2)/(x+7) > 2K\)`

Subcase 2:
`\(((x+1)^2)/(x+7)-3 > 0\)`

Step 4: Choosing the value of
`\(K\)`:

To ensure that the given implication is true for all
`\(x \leq -2\)`, we need to choose a value of
`\(K\)` such that both subcases hold true.

Subcase 1:
`\(3 - ((x+1)^2)/(x+7) > 2K\)`

We want this inequality to hold true for all
`\(x \leq -2\)`. To make it true for the entire range, we should choose the largest possible value for
`\(K\)`.

To do that, we need to find the minimum value of the left side of the inequality, which occurs when
`\(x = -2\)`:

`\[3 - ((-2+1)^2)/(-2+7) = 3 - (1)/(5) = (14)/(5)\]`

So, for this subcase to be true for all
`\(x \leq -2\)`, we should choose
`\(K\)` such that:

`\[(14)/(5) > 2K\]`

Solve for
`\(K\)`:

`\[K < (7)/(10)\]`

Subcase 2:
`\(((x+1)^2)/(x+7)-3 > 0\)`

This inequality is already true for all
`\(x \leq -2\)`, regardless of the value of
`\(K\)`, because the left side is always positive.

Now, we need to ensure that both subcases are satisfied. The maximum value of
`\(K\)` for Subcase 1 is
`\((7)/(10)\)`, and it does not affect Subcase 2.

So, we can choose
`\(K\)` to be any value less than
`\((7)/(10)\)`.

Therefore, one possible choice for
`\(K\)` is
`\(K = (1)/(2)\)`, which is less than
`\((7)/(10)\)`.

The complete question is here:

Answer 2

The value of k for the expression to be true is 5/7.

How k value was determined.

Given

`|x + 4| \leqslant 2 \: | \frac{ {(x + 1)}^(2) }{x + 7} - 3 |`

Simplify and expand, LCM is x + 7

`| \frac{ {x}^(2) + 2x + 1 - 3x - 21 }{x + 7} |`

`| \frac{ {x}^(2) - x - 20}{x + 7} |`

Factorize the numerator

`| ((x + 4)(x - 5))/(x + 7) |`

Meanwhile, the expression should be

`> k |x + 4|`

Compare the factored form

`| (x - 5)/(x + 7) | |x + 4| > k |x + 4|`

Assume x = 0

`| (0 - 5)/(0 + 7) | |x + 4| > k |x + 4|`

Absolute value is

`(5)/(7)`

Therefore, k is < 5/7