Consider the inequality \(|2x – 3| – |x + 2| < 1\).
This question requires you to:
(a) Formulate the piecewise conditions by identifying the critical points where the expressions inside the absolute value change sign.
(b) Solve the inequality on each resulting interval.
(c) Graph the overall solution on a number line, indicating open and closed boundaries appropriately.
(d) Interpret the significance of these solution intervals in a theoretical context, discussing how the constraint boundaries might represent limitations in a modeling scenario.
Question
Formulate the piecewise conditions for the inequality \(|2x – 3| – |x + 2| < 1\) by identifying the critical points where the expressions inside the modulus change sign and writing the inequality explicitly for each interval.
Hint
Hint
Hint
Identify the points where \(2x – 3 = 0\) and \(x + 2 = 0\) to divide the real line.
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Complete Answer
The critical points are \(x = \frac{3}{2}\) from \(2x-3=0\) and \(x = -2\) from \(x+2=0\). Thus, the three intervals are:
1. For \(x < -2\): \(|2x-3| = 3-2x\) and \(|x+2| = -(x+2)\).
2. For \(-2 \le x < \frac{3}{2}\): \(|2x-3| = 3-2x\) and \(|x+2| = x+2\).
3. For \(x \ge \frac{3}{2}\): \(|2x-3| = 2x-3\) and \(|x+2| = x+2\).
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Question
Solve the inequality \(|2x – 3| – |x + 2| < 1\) on each of the intervals formulated in part (a).
Hint
Hint
Hint
Solve the linear inequalities obtained in each interval separately and pay attention to the direction of the inequality.
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For \(x < -2\): the inequality simplifies to \(5 - x < 1\), which gives \(x > 4\) and hence yields no solution in \(x < -2\).
For \(-2 \le x < \frac{3}{2}\): it becomes \(1 - 3x < 1\), leading to \(x > 0\) (thus solution on this interval is \(0 < x < \frac{3}{2}\)).
For \(x \ge \frac{3}{2}\): the inequality becomes \(x - 5 < 1\), giving \(x < 6\) (so solution on this interval is \(\frac{3}{2} \le x < 6\)).
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Question
Graph the overall solution of the inequality on a number line, indicating clearly the status (open or closed) of the endpoints.
Hint
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Combine the solution intervals from part (b) and represent them on a number line.
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The overall solution is \(x \in (0, 6)\). On a number line, this is represented by an open circle at \(x=0\) and an open circle at \(x=6\), with the interval between them shaded.
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Question
Interpret the significance of the solution \(x \in (0, 6)\) in a theoretical context. Discuss how the boundaries of this solution set might represent critical limits in a modeling scenario.
Hint
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Hint
Consider how strict inequalities set boundaries for feasible regions in optimization or constraint problems.
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The solution \(x \in (0, 6)\) indicates that the modeled scenario is only valid between these limits. The exclusion of 0 and 6 signifies that at these boundaries the modeled condition fails to hold (for example, a system may become unstable or unsustainable at these exact values), thereby defining a feasible region for application.
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