Problem 67 Solve and graph. $$ |x-1|&lt... [FREE SOLUTION]

Chapter 8: Problem 67

Solve and graph. $$ |x-1|<4 $$

Short Answer

Expert verified

The solution is $-3 < x < 5$.

Step by step solution

- Understand the Absolute Inequality

The absolute value inequality $|x-1| < 4$ means that the expression inside the absolute value, $x-1$, must lie between -4 and 4.

- Split the Inequality

Rewrite the absolute value inequality without the absolute value sign by creating a double inequality: $-4 < x-1 < 4$.

- Solve the Double Inequality

Solve the inequality $-4 < x-1 < 4$ by adding 1 to all parts of the inequality: $-4 + 1 < x-1 + 1 < 4 + 1$ results in $-3 < x < 5$.

- Graph the Solution

Graph the solution $-3 < x < 5$. This is an open interval from -3 to 5, not including -3 and 5. On a number line, draw an open circle at -3 and 5 and shade the region between them.

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Key Concepts

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

Understanding Inequalities

Inequalities are mathematical statements that express the relative size or order of two objects. An inequality compares two values, showing if one is less, greater, less than or equal to, or greater than or equal to another.
Inequalities use symbols like '<', '>', '鈮�', and '鈮�'. For example, if we say 'x < 5', we mean that x can be any number less than 5. It's like making a mathematical statement about the possible values a number can take.
In the context of absolute value inequalities, these statements indicate a range within which the values can lie.

Graphing Inequalities

Graphing inequalities helps visualize the range of possible solutions. For an inequality like \[x-1 < 4\], we'd first solve it algebraically.
Once solved, the resulting expression tells us the range of x on a number line.
For instance, in the given exercise \[|x-1|<4\], it simplifies to \[-3This visual representation makes it easier to understand which values are part of the solution.

Basic Algebra Approach

Algebra involves using symbols and letters to represent numbers and quantities in formulas and equations. In solving inequalities, we often treat them similarly to equations.
For example, let's re-visit the exercise: \[|x-1|<4\]. Start by understanding that the absolute value means the distance from zero without considering the sign.
To solve it, you split the inequality into two parts: \[-4This results in \[-3

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91影视

Solve and graph. $$ |x-1|<4 $$

Short Answer

Step by step solution

- Understand the Absolute Inequality

- Split the Inequality

- Solve the Double Inequality

- Graph the Solution

Key Concepts

Understanding Inequalities

Graphing Inequalities

Basic Algebra Approach

One App. One Place for Learning.

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