Problem 54 - Precalculus Student Solutions Manual 5th

Question

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

Step-by-Step Solution

Answer

There is no solution.

Step 1: 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

Step 2: 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.

Key Concepts

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.