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Chapter 2: Problem 78

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|5 x-6|=|-5 x+6|$$

Short Answer

Expert verified
The solution is \(x = \frac{6}{5}\). Graphically, \(|5x - 6| = |-5x + 6|\) is equal at \(x = \frac{6}{5}\). For inequalities, explore ranges within other noted values.

Step by step solution

01

Equation Breakdown

First, observe the equation \(|5x - 6| = |-5x + 6|\). The absolute value removes the signs of expressions, so we must consider cases where each side can be either equal or opposite.
02

Case 1: Both expressions are equal

Assume \(5x - 6 = -5x + 6\). Solve this equation for \(x\):\[5x - 6 = -5x + 6\]Add \(5x\) to both sides:\[10x - 6 = 6\]Add 6 to both sides:\[10x = 12\]Divide each side by 10:\[x = \frac{6}{5}\]
03

Case 2: Expressions are opposites

Assume \(5x - 6 = 5x - 6\). The left expression is just \(5x - 6\). Solving this trivial case confirms it for all \(x\) if situations apply, but here:Adding 6 to both sides doesn't make it further specific beyond prior:\[5x = 5x\]
04

Intersection Check via Inequality Conversions

Convert the equation to inequalities. We are effectively checking intersections or distinguished forms:- \(|f(x)| > |g(x)|\) which is \(|5x - 6| > |-5x + 6|\), and- \(|f(x)| < |g(x)|\) which is \(|5x - 6| < |-5x + 6|\).Finding cross over within graphically supporting through calculated values.
05

Analyze Inequality \(|f(x)| > |g(x)|\)

Solve \(|5x - 6| > |-5x + 6|\). We derived:Case 1: \(5x-6 > -5x+6\) simplifies to:\[10x > 12\]Thus, \(x > \frac{6}{5}\) Case 2: \(-5x+6 > 5x-6\) gives:\[-10x > -12\]Flip inequality:\[x < \frac{6}{5}\]Combining, nontrivial part is \(x eq \frac{6}{5}\).
06

Solve Inequality \(|f(x)| < |g(x)|\)

Now solve \(|5x - 6| < |-5x + 6|\):Case 1: \(5x-6 < -5x+6\) simplifies to:\[10x < 12\] gives \(x < \frac{6}{5}\)Case 2: \(-5x + 6 < 5x - 6\) simplifies to the opposite so does not apply. Valid points mostly invert case outputs analyzed last.
07

Graphical Support

Graph both functions \(f(x) = |5x - 6|\) and \(g(x) = |-5x + 6|\). Analyze where both curves intersect or diverge confirming analytical outputs. Both meet at \(\frac{6}{5}\) crossing around values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequalities
In mathematics, inequalities play a vital role. They describe the relationship between two expressions, indicating that one is greater or lesser than the other. In the case of absolute value inequalities, like - \(|5x - 6| > |-5x + 6|\) - \(|5x - 6| < |-5x + 6|\)we actually compare the magnitudes of two mathematical expressions while considering their potential signs. They tell us where one function surpasses or lags behind the other.
To solve these inequalities, break them into case scenarios based on the properties of absolute values.
  • For the condition \(|f(x)| > |g(x)|\), calculate situations where the expression inside \(f(x)\) exceeds \(g(x)\) or lags behind.
  • For \(|f(x)| < |g(x)|\), do the same but inverse the outcomes.
By doing this, you determine ranges where inequalities hold true, allowing visualization on a real number line.
Graphical Analysis
Graphical analysis provides a visual way to understand relationships between functions. By plotting the absolute value functions \(f(x) = |5x - 6|\) and \(g(x) = |-5x + 6|\), you obtain their curve representations,
which highlight intersections and divergences.While visualizing the graphs:
  • Note intersection points, where functions meet at the exact same value.
  • Divergences indicate ranges where one function's magnitude exceeds the other.
Graphical analysis makes it easier to showcase solutions found through analytical means. By examining the graph, such as observing both functions meeting at \(x = \frac{6}{5}\), not only confirms our calculations but provides confidence in solutions linking to inequalities.
Case-Based Analysis
Case-based analysis is crucial when dealing with absolute value equations. It involves assessing possible scenarios for the expressions, especially where multiple absolute values are considered.
In this problem, - First, we acknowledge one case \(5x - 6 = -5x + 6\), simplifying to solve for \(x\).- Another case arises when we reverse sign on one side, observing identical expressions, confirming trivial solutions for any \(x\) that fit a numeric boundary condition.Why perform these analyses?
  • Each case covers unique aspects of the situation, ensuring comprehensive examination of solutions.
  • Handling each condition separately guarantees that all potential solutions are addressed.
Combining results from all cases ensures a full understanding, whether examining intersection points through calculations or inequalities.

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