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Chapter 1: Problem 54

Solve each equation or inequality. $$|-5 x+7|-4<-6$$

Short Answer

Expert verified
There is no solution.

Step by step solution

01

Isolate the absolute value expression

Start by isolating the absolute value expression y adding 4 to both sides of the inequality: ewline ewline ewline ewline ewline ewline ewline ewline |-5x + 7| - 4 < -6 --> |-5x + 7| -4 + 4 < -6 + 4 ewline |-5x + 7| < -2
02

Analyze the absolute value inequality

Consider that the absolute value of a number is always non-negative (greater than or equal to zero). However, the inequality |-5x + 7| < -2 implies that the absolute value expression would be less than a negative number, which is impossible. So, there is no solution.

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Key Concepts

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

Absolute Value Inequalities
Absolute value inequalities involve expressions within absolute value symbols. Absolute value refers to the distance of a number from zero on a number line, disregarding its sign.
It is always non-negative, meaning \(|a| \geq 0\).
For example, \(|-3| = 3\) and \(|5| = 5\).
Inequality Solutions
To solve an inequality involving absolute values, follow these steps:
  • Isolate the absolute value expression if necessary.
  • Consider the nature of absolute values: they are always non-negative.
  • If the inequality asserts that an absolute value is less than a negative number (as in \(|-5x + 7| < -2\)), recognize it's impossible since absolute values cannot be negative.
    Thus, no solution exists in such cases.
Precalculus Problem Solving
When solving precalculus problems, approach them systematically.
Break down the problem into simpler steps: isolate variables, analyze expressions, and apply mathematical principles.
For absolute value inequalities:
  • Ensure the absolute value is isolated.
  • Analyze the resulting expression; if it contradicts fundamental properties (like an absolute value being less than a negative number), there may be no solution.
Developing methodical solving strategies in precalculus can simplify even complex problems substantially.

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Most popular questions from this chapter

Solve each problem. Dr. Tydings has found that, over the years, \(95 \%\) of the babies he has delivered weighed \(x\) pounds, where $$|x-8.2| \leq 1.5.$$ What range of weights corresponds to this inequality?

Solve each equation or inequality. $$|8-3 x|-3=-2$$

Suppose that \(y=5 x+1\) and we want \(y\) to be within 0.002 unit of \(6 .\) For what values of \(x\) will this be true?

Write each statement as an absolute value equation or inequality. \(r\) is no less than 1 unit from 29.

Solve each equation or inequality. $$|4 x+3|-2=-1$$
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