{(x+2)(x-6)<(x+2)(x+1)+4 {2(6x-1)>7(2x-4)

Let's solve the inequalities step-by-step.

1. Solve \((x+2)(x-6) < (x+2)(x+1) + 4\)

First, we'll simplify both sides of the inequality.

• Expand the left side: \[ (x+2)(x-6) = x^2 - 6x + 2x - 12 = x^2 - 4x - 12 \]

• Expand the right side: \[ (x+2)(x+1) + 4 = (x^2 + x + 2x + 2) + 4 = x^2 + 3x + 2 + 4 = x^2 + 3x + 6 \]

Now, we rewrite the inequality: \[ x^2 - 4x - 12 < x^2 + 3x + 6 \]

Next, subtract \(x^2\) from both sides: \[ -4x - 12 < 3x + 6 \]

Now, let's isolate the variable:

1. Add \(4x\) to both sides: \[ -12 < 7x + 6 \]
2. Subtract \(6\) from both sides: \[ -18 < 7x \]
3. Divide by \(7\): \[ -\frac{18}{7} < x \]

So, we have: \[ x > -\frac{18}{7} \]

2. Solve \(2(6x-1) > 7(2x-4)\)

First, expand both sides:

• Left side: \[ 2(6x - 1) = 12x - 2 \]
• Right side: \[ 7(2x - 4) = 14x - 28 \]

Now we rewrite the inequality: \[ 12x - 2 > 14x - 28 \]

Isolate \(x\):

1. Subtract \(12x\) from both sides: \[ -2 > 2x - 28 \]
2. Add \(28\) to both sides: \[ 26 > 2x \]
3. Divide by \(2\): \[ 13 > x \]

So, we have: \[ x < 13 \]

Summary of Solutions

We found two conditions for \(x\):

1. From the first inequality: \(x > -\frac{18}{7}\)
2. From the second inequality: \(x < 13\)

Combining these results, the solution to the system of inequalities is: \[ -\frac{18}{7} < x < 13 \]

This means that \(x\) can take any value in the interval \( \left(-\frac{18}{7}, 13\right) \).