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प्रश्न
Solve the linear inequation, write down the solution set and represent it on the real number line:
\[5(2 - 4x) > 18 - 6x > 22 - 20x, x \in \mathrm{R}\]
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उत्तर
Step 1: Solve the inequation algebraically
The given inequation is \[5(2 - 4x) > 18 - 6x > 22 - 20x, x \in \mathrm{R}\]
This can be broken into two linear inequations:
Part I:
\[5(2 - 4x) > 18 - 6x\]
\[10 - 20x > 18 - 6x\]
\[-20x + 6x > 18 - 10\]
\[-14x > 8\]
Dividing both sides by –14 which reverses the inequality sign:
\[x < -\frac{8}{14}\]
\[x < -\frac{4}{7} (\approx -0.57)\]
Part II:
\[18 - 6x > 22 - 20x\]
\[-6x + 20x > 22 - 18\]
\[14x > 4\]
Dividing both sides by 14:
\[x > \frac{4}{14}\]
\[x > \frac{2}{7} (\approx 0.29)\]
Step 2: Find the intersection of solutions
For the combined inequality to hold, x must satisfy both conditions simultaneously: \[x < -\frac{4}{7} \quad \text{and} \quad x > \frac{2}{7}\]
Since no real number can be simultaneously strictly less than \(-\frac{4}{7}\) and strictly greater than \(\frac{2}{7}\), the two conditions are mutually exclusive: \[\left(-\infty, -\frac{4}{7}\right) \cap \left(\frac{2}{7}, \infty\right) = \phi\]
Solution Set:
\[\text{Solution Set} = \phi \quad \text{or} \quad { } \text{(Empty set)}\]
Representation on the Real Number Line:
Since the solution set contains no elements (empty set), no points or line segments are darkened/shaded on the real number line.
