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Chapter 1: Problem 115

Solve each equation or inequality. $$ |10 x+7|+3<1 $$

Short Answer

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No solution.

Step by step solution

01

Isolate the absolute value expression

Start by isolating the absolute value expression on one side of the inequality. Subtract 3 from both sides. This gives us:\[ |10x + 7| + 3 - 3 < 1 - 3 \]which simplifies to:\[ |10x + 7| < -2 \]
02

Analyze the inequality

Since the absolute value of any real number is always non-negative, it cannot be less than a negative number. Therefore, an inequality of the form \( |expression| < -negative\ _number \) has no solution.

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Key Concepts

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

Solving Inequalities
When solving inequalities, our goal is to find the set of values for the variable that satisfy the given condition.

Absolute value inequalities can sometimes seem confusing, but they follow a straightforward process.
Let's start by isolating the absolute value expression from any constants.
  • In the given problem, we did this by subtracting 3 from both sides of the inequality:
\[ |10x + 7| + 3 - 3 < 1 - 3 \]

This simplified to: \[ |10x + 7| < -2 \]
Interestingly, we noticed that the inequality contains a negative number, which leads us to the next important point regarding absolute values and their properties.
Absolute Value
Absolute value represents the distance of a number from zero on the number line. This distance is always non-negative.
  • For any real number, the absolute value is \[ |a| \] and it is always greater than or equal to zero.


Therefore, an absolute value expression, like \[ |10x + 7| \], cannot be less than a negative number.
It simply doesn鈥檛 make logical sense, because distances can't be negative.
Non-Negative Property
The non-negative property of absolute value is crucial to understanding why the inequality \[ |10x + 7| < -2 \] has no solution.

When an absolute value is compared to a negative number, such as being smaller than a negative value, it results in an impossibility.
This is due to the nature of absolute values always being zero or positive.
  • To summarize, if you encounter something like \[ |expression| < -negative number \], know immediately that there is no solution.


The non-negative property helps in shortcutting the problem-solving steps for absolute value inequalities, making it clear and concise.

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