To solve the inequality [tex]\(-\frac{2}{3} x > -2\)[/tex], let's follow these steps:
1. Eliminate the Fraction:
To eliminate the fraction, we can multiply both sides of the inequality by the reciprocal of [tex]\(-\frac{2}{3}\)[/tex]. The reciprocal of [tex]\(-\frac{2}{3}\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex]. Remember, when we multiply or divide by a negative number, we must reverse the inequality sign.
[tex]\[
-\frac{3}{2} \cdot \left( -\frac{2}{3} x \right) < -\frac{3}{2} \cdot (-2)
\][/tex]
2. Simplify Both Sides:
On the left side, the [tex]\(-\frac{3}{2}\)[/tex] and the [tex]\(-\frac{2}{3}\)[/tex] will cancel each other out, leaving [tex]\( x \)[/tex]:
[tex]\[
x < -\frac{3}{2} \cdot (-2)
\][/tex]
Now calculate the right side:
[tex]\[
-\frac{3}{2} \times (-2) = (-3 \times -2) / 2 = 6 / 2 = 3
\][/tex]
Therefore, the inequality simplifies to:
[tex]\[
x < 3
\][/tex]
3. Solution in Interval Notation:
The result [tex]\( x < 3 \)[/tex] means that [tex]\( x \)[/tex] can be any number less than 3. In interval notation, this is written as:
[tex]\[
(-\infty, 3)
\][/tex]
4. Conclusion:
The solution to the inequality [tex]\(-\frac{2}{3} x > -2\)[/tex] is that [tex]\( x \)[/tex] falls in the interval:
[tex]\[
\boxed{(-\infty, 3)}
\][/tex]
