https://file.aiquickdraw.com/mathbot/file/17270234541945fitx7ov.jpg

Cari himpunan penyelesaian dari ketidaksamaan berikut.

The problem from the image is:

\[
\frac{x + 2}{4 - 2x} \geq 1 - x
\]

We are tasked with finding the solution set for this inequality. Let's go through the steps to solve this.

### Step 1: Simplify the inequality

First, rewrite the inequality as:

\[
\frac{x + 2}{4 - 2x} \geq 1 - x
\]

To eliminate the fraction, multiply both sides of the inequality by \( 4 - 2x \), but we need to consider two cases, since the sign of \( 4 - 2x \) changes depending on whether \( x \) is greater or less than \( 2 \).

### Case 1: \( 4 - 2x > 0 \) (i.e., \( x < 2 \))

1. Multiply both sides by \( 4 - 2x \):
\[
x + 2 \geq (1 - x)(4 - 2x)
\]
2. Expand the right side:
\[
x + 2 \geq (4 - 2x - 4x + 2x^2)
\]
Simplifying:
\[
x + 2 \geq 2x^2 - 6x + 4
\]
3. Bring all terms to one side:
\[
0 \geq 2x^2 - 7x + 2
\]
or equivalently:
\[
2x^2 - 7x + 2 \leq 0
\]

Now, solve the quadratic inequality \( 2x^2 - 7x + 2 \leq 0 \).

The discriminant of this quadratic is:
\[
\Delta = (-7)^2 - 4(2)(2) = 49 - 16 = 33
\]
So, the roots are:
\[
x = \frac{-(-7) \pm \sqrt{33}}{2(2)} = \frac{7 \pm \sqrt{33}}{4}
\]
These roots are approximately \( x_1 = 0.21 \) and \( x_2 = 1.79 \).

Since the quadratic opens upwards, the solution to \( 2x^2 - 7x + 2 \leq 0 \) lies between the roots:
\[
0.21 \leq x \leq 1.79
\]

### Case 2: \( 4 - 2x < 0 \) (i.e., \( x > 2 \))

In this case, multiplying by \( 4 - 2x \) flips the inequality sign:
\[
x + 2 \leq (1 - x)(4 - 2x)
\]
Following the same steps as before, this case simplifies in the same way and leads to no solutions in the interval \( x > 2 \).

### Conclusion:

The solution set for the inequality is approximately \( 0.21 \leq x \leq 1.79 \), within the region where \( x < 2 \).

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions:
1. How do we solve quadratic inequalities step by step?
2. Why do we have to consider cases when multiplying by a variable expression?
3. What is the discriminant, and how does it help solve quadratic equations?
4. Can an inequality have no solution, and how do we determine that?
5. How do you graphically represent the solution to a quadratic inequality?

**Tip**: Always check the sign of the expression you're multiplying by when solving inequalities, as it affects the direction of the inequality.

Cari himpunan penyelesaian dari pertidaksamaan berikut: (x + 2) / (4 - 2x) ≥ 1 - x

Learn how to solve the rational inequality (x + 2) / (4 - 2x) ≥ 1 - x. This detailed step-by-step solution involves solving quadratic inequalities using the discriminant and quadratic formula. Suitable for students in Grades 10-12.

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