Question

Solve the linear inequation, write down the solution set and represent it on the real number line:

\[5(2 - 4x) > 18 - 6x > 22 - 20x, x \in \mathrm{R}\]

Answer 1

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.