Question

1) x²-3-4<0 2) x²-3x-4>0 3) 2x²+3x-5>0 4) -6x²+6x+36>0

Answer 1

1) x²-3-4<0;(х-√7)(х+√7)<0;

____-√7________√7___________

  +                -                +

х∈(-√7;√7)

второе решение на случай, если при втором коэффициенте потерян х.   x²-3х-4<0; по Виету корни левой части -1 и 4.

____-1_______4___________

   +              -                 +

х∈(-1;4)

2) x²-3x-4>0 уже решил. как второй вариант первого. только здесь другой знак. поэтому ответ х∈(-∞;-2.5)∪(1;+∞)

Решение

x²-3х-4>0; по Виету корни левой части -1 и 4.

____-1_______4___________

   +              -                 +

Ответ х∈(-∞;-2.5)∪(1;+∞)

3) 2x²+3x-5>0 , по Виету корни левой части 1 и -2.5

_____-2.5______1_________

+                   -              +

х∈(-∞;-2.5)∪(1;+∞)

4) -6x²+6x+36>0

-6*(x²-x-6)>0; (x²-x-6)<0;

По Виету x=-2; x=3.

______-2_____3__________

  +               -          +

х∈(-2;3)

Answer 2

Let's solve each inequality one by one:

1. x² - 3x - 4 < 0:

To solve this quadratic inequality, we need to find the values of x that make the expression less than zero.

Step 1: Factorize the quadratic equation: (x - 4)(x + 1) < 0

Step 2: Determine the sign of each factor: For (x - 4) = 0, x = 4 For (x + 1) = 0, x = -1

Step 3: Create a sign chart to determine the solution: We have three intervals on the number line: (-∞, -1), (-1, 4), and (4, ∞).

Test a value from each interval to determine the sign: For x = -2: (-2 - 4)(-2 + 1) < 0 ⟹ (-6)(-1) < 0 ⟹ 6 > 0 (true) For x = 0: (0 - 4)(0 + 1) < 0 ⟹ (-4)(1) < 0 ⟹ -4 < 0 (true) For x = 5: (5 - 4)(5 + 1) < 0 ⟹ (1)(6) < 0 ⟹ 6 < 0 (false)

Step 4: Analyzing the sign chart: From the sign chart, we observe that the inequality is true in the interval (-1, 4).

Therefore, the solution to the inequality x² - 3x - 4 < 0 is -1 < x < 4.

2. x² - 3x - 4 > 0:

Similar to the previous inequality, we'll solve this quadratic inequality by finding the values of x that make the expression greater than zero.

Step 1: Factorize the quadratic equation: (x - 4)(x + 1) > 0

Step 2: Determine the sign of each factor: For (x - 4) = 0, x = 4 For (x + 1) = 0, x = -1

Step 3: Create a sign chart to determine the solution: The intervals on the number line are (-∞, -1), (-1, 4), and (4, ∞).

Test a value from each interval to determine the sign: For x = -2: (-2 - 4)(-2 + 1) > 0 ⟹ (-6)(-1) > 0 ⟹ 6 > 0 (true) For x = 0: (0 - 4)(0 + 1) > 0 ⟹ (-4)(1) > 0 ⟹ -4 > 0 (false) For x = 5: (5 - 4)(5 + 1) > 0 ⟹ (1)(6) > 0 ⟹ 6 > 0 (true)

Step 4: Analyzing the sign chart: From the sign chart, we observe that the inequality is true in the intervals (-∞, -1) and (4, ∞).

Therefore, the solution to the inequality x² - 3x - 4 > 0 is x < -1 or x > 4.

3. 2x² + 3x - 5 > 0:

This is another quadratic inequality. Let's solve it:

Step 1: Factorize the quadratic equation: 2x² + 3x - 5 > 0

Since the quadratic equation cannot be easily factored, we can use the quadratic formula to find