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Chapter 4: Problem 88

Let \(f(x)=|2-9 x| .\) Find all \(x\) for which \(f(x) \geq 25\)

Short Answer

Expert verified
x \( \leq -\frac{23}{9}\) or x \( \geq 3\)

Step by step solution

01

Understand the absolute value function

The absolute value function \(f(x)=|2-9x|\) ensures that the expression inside is always non-negative. This means \(f(x)=2-9x\) if \(2-9x \geq 0\) and \(f(x)=-(2-9x)\) if \(2-9x < 0\).
02

Create inequalities for each condition

First, solve \(2-9x \geq 25\): \[ 2-9x \geq 25 \rightarrow -9x \geq 23 \rightarrow x \leq -\frac{23}{9} \] Next, solve \- \left(2-9x \right) \geq 25: \[ 2-9x \leq -25 \rightarrow -9x \leq -27 \rightarrow x \geq 3 \]
03

Combine solutions

From the inequalities above, we see that \(x \leq -\frac{23}{9}\) and \(x \geq 3\). Therefore, the solutions are \(x \leq -\frac{23}{9}\) or \(x \geq 3\).

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Key Concepts

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

absolute value functions
An absolute value function makes sure its output is always non-negative. For any real number inside the absolute value brackets, only the positive value is considered. So, if we have \( f(x) = |2-9x| \), it's telling us how far \(2-9x\) is from zero on the number line. To handle absolute value expressions, you break them into two cases:
  • If the inside expression is already non-negative, we leave it as it is: \( f(x) = 2 - 9x \).
  • If the inside expression is negative, we change its sign to make it positive: \( f(x) = -(2 - 9x) \).
This split into two cases is essential to solve absolute value equations and inequalities.
inequalities
Inequalities show that one side is larger or smaller than the other, without being exactly equal. For the exercise's inequality \( f(x) \geq 25 \), we split based on absolute value conditions:
  • If \( 2 - 9x \geq 0 \), then \( 2 - 9x \geq 25 \), simplifying to \( x \leq -\frac{23}{9} \).
  • If \( 2 - 9x < 0 \), then \( - (2 - 9x) \geq 25 \), meaning \( 2 - 9x \leq -25 \) and simplifying to \( x \geq 3 \).
By solving these separately, we establish boundaries for the possible \( x \) values. Remember, solving inequalities involves reversing inequality signs when multiplying or dividing by negative quantities.
solution sets
A solution set includes all values that satisfy the given equation or inequality. In this problem, we combined results from both cases of the absolute value inequality \( f(x) \geq 25 \). We found:
  • When \( 2 - 9x \geq 25 \), \( x \leq -\frac{23}{9} \).
  • When \( 2 - 9x \leq -25 \), \( x \geq 3 \).
Thus, the solution set consists of all \( x \) values where
\( x \leq -\frac{23}{9} \) or \( x \geq 3 \).
This tells us that any number to the left of \( -\frac{23}{9} \) on the number line or to the right of 3 works in the given inequality, giving us a clear picture of our solution set.

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