Problem 2 - Algebra 2 and Trigonometry

Question

Maria said that since the solution of the inequality \(|2 x-5| < 3\) can be found by using \(-3 < 2 x-5 < 3,\) then the solution of the inequality \(|2 x-5| > 3\) can be found by using \(-3 > 2 x-5 > 3 .\) Do you agree with Maria? Explain why or why not.

Step-by-Step Solution

Answer

No, Maria is incorrect. The correct solution is \(x < 1\) or \(x > 4\).

Step 1: Understanding Absolute Value Inequalities

First, let's review the properties of absolute value inequalities. For \(|a| < b\), the solution is given by \(-b < a < b\). This is because the expression within the absolute value is bounded by \(b\) from both directions.

Step 2: Application of the Absolute Value Rule

Maria is correct in stating that for \(|2x - 5| < 3\), the solution is \(-3 < 2x - 5 < 3\). We apply the rule for absolute value inequalities to split it into a compound inequality.

Step 3: Analyzing the Logic for Greater Than Inequality

Now let's consider the inequality \(|2x - 5| > 3\). The solution set for this inequality is different. It requires that either \(2x - 5 > 3\) or \(2x - 5 < -3\). This is due to the fact that the expression within the absolute value is greater than \(3\) in magnitude.

Step 4: Contradiction in Maria's Approach

Maria suggested using \(-3 > 2x - 5 > 3\) for \(|2x - 5| > 3\). This doesn't make sense because it implies the expression has values both greater than 3 and less than -3 simultaneously, which can't happen.

Step 5: Correct Method for Solving \(|a| > b\)

The correct approach for solving \(|a| > b\) is to split the inequality into two parts: \(a > b\) or \(a < -b\). These two possibilities cover all values for which the magnitude of \(a\) is greater than \(b\).

Step 6: Applying Correct Method to \(|2x - 5| > 3\)

According to the rule, we split \(|2x - 5| > 3\) into \(2x - 5 > 3\) and \(2x - 5 < -3\). Solving these gives the intervals \(2x > 8\), thus \(x > 4\), and \(2x < 2\), thus \(x < 1\). Therefore, the solution is \(x < 1\) or \(x > 4\).

Key Concepts

Understanding Compound Inequalities

Compound inequalities involve expressing two inequalities simultaneously which describe a range of solutions for a single variable.

The compound inequalities are often written in the form of

• a conjunction, such as \(a < x < b\), which requires a variable to satisfy both conditions simultaneously
• an intersection of intervals
• a disjunction, such as \(x < a\) or \(x > b\), which means a variable can satisfy either condition

Particularly, when dealing with absolute value inequalities, compound inequalities help carefully craft conditions to match the requirements.

For example, when \(|2x - 5| < 3\), it translates to \(-3 < 2x - 5 < 3\), which further makes solving the inequality straightforward and ensures the parameter is correctly bounded.

Solving Inequalities: Achieving Correct Solutions

Solving inequalities, especially absolute value inequalities, involves making careful considerations about the range a variable can take.
The inequality \(|a| < b\) means the expression inside is bounded between -\(b\) and \(b\). This gives a solution of the form \(-b < a < b\), effectively capturing the context where the absolute deviation is within limit.To solve the inequality \(|a| > b\), the scenario shifts. Here, you have two separate conditions:

• either the expression is greater than \(b\)
• or less than -\(b\)

This condition generates a valid solution by covering situations outside the specified range, resulting in expressions of the form \(a > b\) or \(a < -b\).
This understanding helps avoid incorrect solutions like Maria suggested, where she mistakenly combined the elements into an impossible scenario of \(-3 > 2x-5 > 3\).

Mastering Magnitude Comparisons in Inequalities

Magnitude comparisons often make dealing with inequalities involving absolute values more intuitive and visible.
The key concept here is understanding that absolute value inequalities revolve around measuring the distance from zero, which makes them naturally suited for addressing conditions about greater or lesser magnitudes. For an inequality such as \(|2x - 5| > 3\), you're asked to consider when the magnitude of \(2x - 5\) exceeds 3.

• This implies an expression further removed from 0 by more than 3 on either side of the number line.
• A careful setting of conditions
• ensures solution captures valid intervals: \(2x - 5 > 3\)
• or \(2x - 5 < -3\), hence focusing more on the magnitude shift.

Understanding and applying magnitude comparisons reveal the true nature of absolute value inequalities, providing clarity and precision in solving them.