Question

Изобразите на числовой оси решение системы не равинств {2x^2-5x-7>=0 {X>=0

Answer 1

Реши первое по дискриминанту и получишь вот такую числовую ось)

Answer 2

Solving the Inequality System on a Number Line

To solve the system of inequalities {2x^2-5x-7>=0, x>=0, we can first find the solutions to each inequality separately and then represent the combined solution on a number line.

Solving 2x^2-5x-7>=0:

The solutions to the inequality 2x^2-5x-7>=0 can be found by determining the values of x for which the quadratic expression is greater than or equal to zero. This can be done by finding the roots of the quadratic equation 2x^2-5x-7=0 and then testing the intervals between the roots.

Using the quadratic formula, the roots of the quadratic equation 2x^2-5x-7=0 are: x = (-(-5) ± √((-5)^2 - 4*2*(-7))) / (2*2) x = (5 ± √(25 + 56)) / 4 x = (5 ± √81) / 4 x = (5 ± 9) / 4

So, the roots are x = (5 + 9) / 4 and x = (5 - 9) / 4: x1 = 14 / 4 = 3.5 x2 = -4 / 4 = -1

This means the quadratic expression changes sign at x = -1 and x = 3.5. We can test the intervals (-∞, -1), (-1, 3.5), and (3.5, +∞) to determine where the expression is greater than or equal to zero.

Testing the interval (-∞, -1): Choose x = -2: 2(-2)^2 - 5(-2) - 7 = 2*4 + 10 - 7 = 8 + 10 - 7 = 11 > 0 So, the expression is greater than zero in this interval.

Testing the interval (-1, 3.5): Choose x = 0: 2(0)^2 - 5(0) - 7 = -7 < 0 So, the expression is less than zero in this interval.

Testing the interval (3.5, +∞): Choose x = 4: 2(4)^2 - 5(4) - 7 = 2*16 - 20 - 7 = 32 - 20 - 7 = 12 > 0 So, the expression is greater than zero in this interval.

Therefore, the solution to the inequality 2x^2-5x-7>=0 is x<=-1 or x>=3.5.

Solving x>=0:

The solution to the inequality x>=0 is simply x>=0.

Representing the Solution on a Number Line:

Now, we can represent the combined solution on a number line. The solution to the system of inequalities {2x^2-5x-7>=0, x>=0 is the intersection of the solutions to each individual inequality.

The solution to the system is x>=0 and x<=-1 or x>=3.5. This can be represented on a number line as follows:

  -1        0       3.5
---|--------|-------|---|---
   x<=-1   x>=0   x>=3.5

The shaded regions on the number line represent the values of x that satisfy the system of inequalities.

I hope this helps! If you have any further questions, feel free to ask.