Question

6-4x/x+0,5>0 решить неравенство

Answer 1

$\frac{6-4x}{x+0,5} \ \textgreater \ 0\\\\ \frac{-2(2x-3)}{(2x+1):2} \ \textgreater \ 0\\\\ \frac{4(2x-3)}{2x+1} \ \textless \ 0\\\\+++(\frac{1}{2})---(\frac{3}{2})+++\\\\x\in (\frac{1}{2};\frac{3}{2})$

Answer 2

To solve the inequality (6 - 4x)/(x + 0.5) > 0, we can use the following steps:

1. Find the critical values of x that make the numerator or denominator of the expression equal to zero. These values are -1.5 and 1.5, respectively.

2. Use these critical values to divide the number line into three intervals: x < -1.5, -1.5 < x < 1.5, and x > 1.5.

3. Test a value in each interval to determine whether the expression is positive or negative in that interval.

4. Write the solution set as the union of the intervals where the expression is positive.

Step 1: Find the critical values

The numerator of the expression is 6 - 4x, which equals zero when x = 1.5. The denominator is x + 0.5, which equals zero when x = -0.5.

So, the critical values are x = -0.5 and x = 1.5.

Step 2: Divide the number line into intervals

We now have three intervals to consider: x < -0.5, -0.5 < x < 1.5, and x > 1.5.

Step 3: Test a value in each interval

To test whether the expression is positive or negative in each interval, we can choose a test value in each interval and substitute it into the expression. A convenient test value in each interval is the midpoint of the interval.

For x < -0.5, we can choose x = -1. The expression becomes:

(6 - 4(-1))/(-1 + 0.5) = 10/(-0.5) < 0

Therefore, the expression is negative in the interval x < -0.5.

For -0.5 < x < 1.5, we can choose x = 0.5. The expression becomes:

(6 - 4(0.5))/(0.5 + 0.5) = 4 > 0

Therefore, the expression is positive in the interval -0.5 < x < 1.5.

For x > 1.5, we can choose x = 2. The expression becomes:

(6 - 4(2))/(2 + 0.5) = -2.4 < 0

Therefore, the expression is negative in the interval x > 1.5.

Step 4: Write the solution set

The inequality is satisfied when the expression is greater than zero. Therefore, the solution set is the union of the intervals where the expression is positive, which is:

-0.5 < x < 1.5

So the solution to the inequality (6 - 4x)/(x + 0.5) > 0 is the interval (-0.5, 1.5).