Question

6) (3x - 1)-(x - 1)2 > 4(x + 4);

Answer 1

Ответ:

Во вложении

Объяснение:

Answer 2

To solve the inequality (3x - 1) - (x - 1)^2 > 4(x + 4), you'll want to simplify it step by step.

1. Start by expanding the squared term (x - 1)^2: (3x - 1) - (x^2 - 2x + 1) > 4(x + 4)

2. Distribute the 4 on the right side: 3x - 1 - x^2 + 2x - 1 > 4x + 16

3. Combine like terms on both sides of the inequality: (3x + 2x) - 1 - 1 - x^2 > 4x + 16

4. Combine like terms further: 5x - 2 - x^2 > 4x + 16

5. Move all terms to one side to set the inequality to zero: 5x - 4x - x^2 - 2 - 16 > 0

6. Simplify the left side: x - x^2 - 18 > 0

7. Rearrange the terms and factor out -1: -x^2 + x - 18 > 0

Now, to solve the quadratic inequality, you can use the method of intervals or a sign chart. I'll use a sign chart.

1. Find the critical points by setting the expression equal to 0: -x^2 + x - 18 = 0

You can solve for x by factoring or using the quadratic formula. In this case, I'll use the quadratic formula: x = [ -1 ± √(1 - 4(-1)(-18)) ] / (2(-1)) x = [ -1 ± √(1 + 72) ] / -2 x = [ -1 ± √73 ] / -2

So, the critical points are: x₁ = (-1 + √73) / -2 x₂ = (-1 - √73) / -2

2. Create a sign chart using these critical points. Test a value in each interval: -∞ -1.8019... -1 + √73/2 -1 - √73/2 +∞

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   +       -        +        +       -
   

3. Now, determine the sign of the expression in each interval. The expression is -x^2 + x - 18: -∞: Negative (-1.8019..., -1 + √73/2): Positive (-1 - √73/2, -1 + √73/2): Positive (-1.8019..., +∞): Negative

4. The inequality is satisfied when the expression is greater than 0, so we want the intervals where it is positive: (-1.8019..., -1 + √73/2) U (-1 - √73/2, -1 + √73/2)

So, the solution to the inequality is: x ∈ (-1.8019..., -1 + √73/2) U (-1 - √73/2, -1 + √73/2)