Question

Решить неравенство (x+3)/(x-2)>0 и найти область значений

Answer 1

(x+3)/(x-2)>0 |*(x-2)
(x+3)*(x-2)>0*(x-2)
x²-2x+3x-6>0
x²+x-6>0
a=1 b=1 c=-6
x₁=(-b-√(b²-4ac))/2a=
=(-1-√(1+24))/(2*1)=
=(-1-√25)/2=
=(-1-5)/2=
=-6/2=
=-3

x₂=(-b+√(b²-4ac))/2a=
=(-1+√(1+24))/(2*1)=
=(-1+√25)/2=
=(-1+5)/2=
=4/2=
=2

Answer 2

Solving the Inequality

To solve the inequality (x + 3)/(x - 2) > 0 and find its domain, we can follow these steps:

1. Find the critical points by setting the numerator and denominator equal to zero and solving for x. 2. Use these critical points to create intervals on the number line. 3. Test each interval to determine where the inequality is satisfied.

Critical Points

The critical points for the inequality (x + 3)/(x - 2) > 0 are found by setting the numerator and denominator equal to zero and solving for x. The critical points are the values of x that make the inequality undefined or equal to zero.

The critical points are: - x = -3 (from the numerator) - x = 2 (from the denominator)

Intervals on the Number Line

We can use the critical points to create intervals on the number line. The intervals are: 1. x < -3 2. -3 < x < 2 3. x > 2

Testing the Intervals

We can now test each interval to determine where the inequality (x + 3)/(x - 2) > 0 is satisfied.

1. For x < -3: - Let's choose x = -4. Substituting x = -4 into the inequality gives: (-4 + 3)/(-4 - 2) = -1/-6 < 0. This interval does not satisfy the inequality.

2. For -3 < x < 2: - Let's choose x = 0. Substituting x = 0 into the inequality gives: (0 + 3)/(0 - 2) = 3/-2 < 0. This interval does not satisfy the inequality.

3. For x > 2: - Let's choose x = 3. Substituting x = 3 into the inequality gives: (3 + 3)/(3 - 2) = 6/1 > 0. This interval satisfies the inequality.

Conclusion

The solution to the inequality (x + 3)/(x - 2) > 0 is: x > 2

Domain of the Inequality

The domain of the inequality (x + 3)/(x - 2) is all real numbers except x = 2, as division by zero is undefined.

Therefore, the domain of the inequality is: x ∈ (-∞, 2) U (2, +∞)