To solve the inequality

$\left\vert 2x - 7 \right\vert > \left\vert 2 - 3x \right\vert$

we need to consider different cases based on the points where the expressions inside the absolute values change sign, which occurs when the expressions are equal to zero. These points will divide the number line into intervals that we can analyze separately.

Step 1: Identify the critical points

The critical points are found by setting each expression inside the absolute values equal to zero.

1. $2x - 7 = 0$ $x = \frac{7}{2} = 3.5$

2. $2 - 3x = 0$ $x = \frac{2}{3} \approx 0.67$

So, the critical points are $x = \frac{2}{3}$ and $x = \frac{7}{2}$.

Step 2: Analyze the intervals

The critical points divide the number line into three intervals:

• $(-\infty, \frac{2}{3})$
• $\left(\frac{2}{3}, \frac{7}{2}\right)$
• $\left(\frac{7}{2}, \infty\right)$

We'll evaluate the inequality on each interval.

Interval 1: $(-\infty, \frac{2}{3})$

In this interval:

• $2x - 7 < 0$ (negative)
• $2 - 3x > 0$ (positive)

Thus, the inequality becomes: $-(2x - 7) > 2 - 3x$ Simplify: $-2x + 7 > 2 - 3x$ Add $3x$ to both sides: $x + 7 > 2$ Subtract 7 from both sides: $x > -5$ In this interval, $x > -5$. Since this entire interval is greater than $-5$, this interval is valid for the solution.

Interval 2: $\left(\frac{2}{3}, \frac{7}{2}\right)$

In this interval:

• $2x - 7 < 0$ (negative)
• $2 - 3x < 0$ (negative)

Thus, the inequality becomes: $-(2x - 7) > -(2 - 3x)$ Simplify: $-2x + 7 > -2 + 3x$ Add $2x$ to both sides: $7 > -2 + 5x$ Add 2 to both sides: $9 > 5x$ Divide by 5: $x < \frac{9}{5} = 1.8$ So in this interval, $x < 1.8$.

Interval 3: $\left(\frac{7}{2}, \infty\right)$

In this interval:

• $2x - 7 > 0$ (positive)
• $2 - 3x < 0$ (negative)

Thus, the inequality becomes: $2x - 7 > -(2 - 3x)$ Simplify: $2x - 7 > -2 + 3x$ Subtract $2x$ from both sides: $-7 > -2 + x$ Add 2 to both sides: $-5 > x$ This simplifies to $x < -5$, but since $x > \frac{7}{2}$ in this interval, there are no solutions in this interval.

Step 3: Combine the solutions

The solution for $x$ from the valid intervals is: $x \in (-5, \frac{2}{3}) \cup \left(\frac{2}{3}, 1.8\right)$

Since $x = \frac{2}{3}$ doesn't satisfy the original inequality, it is excluded.

Final Answer:

$x \in (-5, 1.8)$

Let me know if you want any more details or have any questions!

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Here are some related questions:

1. How do you solve absolute value inequalities in general?
2. What are some real-world applications of absolute value inequalities?
3. How does graphing help in solving absolute value inequalities?
4. Can you explain the difference between "and" and "or" in compound inequalities?
5. What is the significance of critical points in analyzing inequalities?

Tip: Always check the intervals defined by the critical points separately when dealing with absolute value inequalities.