Question

Log3+x (3)+log3+x (x^2)<=log3+x (x+4).

Answer 1

ОДЗ :

1) x > 0

2) x + 4 > 0     ⇒    x > - 4

3) 3 + x > 0     ⇒    x > - 3

4) 3 + x ≠ 1      ⇒    x ≠ - 2

Окончательно : x > 0

$log_{3+x}3+log_{3+x}x^{2} \leq log_{3+x}(x+4)\\\\log_{3+x}3x^{2}\leq log_{3+x}(x+4)\\\\3x^{2}\leq x+4\\\\3x^{2} -x-4\leq 0\\\\3(x-1\frac{1}{3})(x+3)\leq 0\\\\(x-1\frac{1}{3})(x+3)\leq 0$

         +                            -                                +

___________(- 3)___________[1 1/3]___________

                        ////////////////////////////////

x ∈ ( - 3 ; 1 1/3]

С учётом, ОДЗ окончательный ответ : x ∈ (0 ; 1 1/3]

Answer 2

To solve the given inequality:

log3+x (3) + log3+x (x^2) <= log3+x (x+4)

We can use the properties of logarithms to simplify the expression.

1. Combine the logarithms using the product rule of logarithms:

log3+x (3 * x^2) <= log3+x (x+4)

2. Apply the quotient rule of logarithms:

log3+x (3x^2 / (x+4)) <= 0

3. Convert the logarithmic inequality to an exponential inequality:

3x^2 / (x+4) <= (3+x)^0

Note: (3+x)^0 simplifies to 1, so the inequality becomes:

3x^2 / (x+4) <= 1

4. Multiply both sides by (x+4) to eliminate the denominator (assuming x + 4 > 0):

3x^2 <= x+4

5. Rearrange the equation to get a quadratic inequality:

3x^2 - x - 4 <= 0

6. Solve the quadratic inequality by factoring or using the quadratic formula:

3x^2 - x - 4 = (3x + 4)(x - 1) <= 0

To find the values of x that satisfy the inequality, we need to determine when the expression (3x + 4)(x - 1) is less than or equal to zero.

7. Set each factor equal to zero and solve:

3x + 4 = 0 => x = -4/3 x - 1 = 0 => x = 1

8. Plot these points on a number line:

-----------o---o-----------

-4/3 1

9. Test intervals:

• Test a value in the interval (-∞, -4/3): Let x = -2 Substitute this value into the inequality: (3(-2) + 4)(-2 - 1) = (-2)(-3) = 6 > 0 The inequality is not satisfied in this interval.

• Test a value in the interval (-4/3, 1): Let x = 0 Substitute this value into the inequality: (3(0) + 4)(0 - 1) = (4)(-1) = -4 <= 0 The inequality is satisfied in this interval.

• Test a value in the interval (1, +∞): Let x = 2 Substitute this value into the inequality: (3(2) + 4)(2 - 1) = (10)(1) = 10 > 0 The inequality is not satisfied in this interval.

10. Determine the solution:

The solution to the inequality 3x^2 - x - 4 <= 0 is x ∈ (-4/3, 1].

Note: It's essential to double-check the solution by plugging in values from the intervals to ensure the inequality holds true.