To solve the inequality $-3|2x-1|+7<4$, let's go step by step.

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Step 1: Simplify the inequality

$-3|2x-1| + 7 < 4$ Subtract 7 from both sides: $-3|2x-1| < -3$ Divide through by $-3$, remembering to reverse the inequality sign since we are dividing by a negative number: $|2x-1| > 1$

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Step 2: Solve the absolute value inequality

The inequality $|2x-1| > 1$ splits into two cases:

1. $2x-1 > 1$
2. $2x-1 < -1$

Case 1: $2x-1 > 1$

$2x > 2 \implies x > 1$

Case 2: $2x-1 < -1$

$2x < 0 \implies x < 0$

Thus, the solution to the inequality is: $x < 0 \quad \text{or} \quad x > 1$

In interval notation: $(-\infty, 0) \cup (1, \infty)$

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Step 3: Graphing the inequality

The graph of the solution involves:

1. Open circles at $x=0$ and $x=1$, because the inequality is strict ($>$).
2. Shading to the left of $x=0$ and to the right of $x=1$.

Let me create the graph for you.import matplotlib.pyplot as plt import numpy as np

Define the range for x values

x = np.linspace(-2, 3, 500)

Plot the critical points and shading

plt.figure(figsize=(8, 4)) plt.axvline(0, color='red', linestyle='--', label='x = 0 (boundary)') plt.axvline(1, color='blue', linestyle='--', label='x = 1 (boundary)')

Highlight the solution regions

plt.fill_betweenx([0, 1], -2, 0, color='green', alpha=0.3, label='x < 0') plt.fill_betweenx([0, 1], 1, 3, color='yellow', alpha=0.3, label='x > 1')

Add labels and legend

plt.axhline(0, color='black', linewidth=0.8) plt.title('Graph of Solution: $x < 0$ or $x > 1$') plt.xlabel('x') plt.ylabel('y (arbitrary scale)') plt.legend() plt.grid()

Show the graph

plt.show() <<ImageDisplayed>>successHere is the graph of the solution:

• The red dashed line represents the boundary at $x = 0$.
• The blue dashed line represents the boundary at $x = 1$.
• The shaded regions indicate where the inequality holds: $x < 0$ (green) and $x > 1$ (yellow).

Would you like any additional details or explanations?

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Related Questions:

1. How do absolute value inequalities differ when the sign is $\leq$ or $=$?
2. Can you explain how to handle compound inequalities with absolute values?
3. What are other methods to graph inequalities?
4. How does this inequality change if the constant outside the absolute value is positive?
5. Could you show step-by-step verification of the solution?

Tip:

Always test points within each interval to verify the solution of absolute value inequalities.