Which inequality represents the solution of |x - 4| < 3? A) 1 < x < 3 B) 1 < x < 3 or x > 7 C) 1 < x < 7 D) -3 < x < 7
Understand the Problem
The question is asking us to solve the absolute value inequality $|x - 4| < 3$ and determine which of the provided options correctly represents this solution. To solve this kind of inequality, we need to consider the definition of absolute value and break it down into two separate inequalities.
Answer
$1 < x < 7$
Answer for screen readers
The solution to the inequality $|x - 4| < 3$ is $1 < x < 7$.
Steps to Solve
- Break down the absolute value inequality
The inequality $|x - 4| < 3$ means that the expression inside the absolute value, $x - 4$, is less than 3 and greater than -3. This can be rewritten as two separate inequalities:
$$ -3 < x - 4 < 3 $$
- Solve the left inequality
Starting with the left side:
$$ -3 < x - 4 $$
To isolate $x$, add 4 to both sides:
$$ -3 + 4 < x $$
Which simplifies to:
$$ 1 < x $$
- Solve the right inequality
Now, we will solve the right side:
$$ x - 4 < 3 $$
Again, add 4 to both sides to isolate $x$:
$$ x < 3 + 4 $$
This simplifies to:
$$ x < 7 $$
- Combine the results
Now, we combine both inequalities obtained from steps 2 and 3:
$$ 1 < x < 7 $$
This tells us that $x$ must be between 1 and 7, not including the endpoints.
The solution to the inequality $|x - 4| < 3$ is $1 < x < 7$.
More Information
This inequality indicates that the values of $x$ that satisfy the original condition are those between 1 and 7. The endpoints, 1 and 7, are not included because of the strict inequality (<).
Tips
One common mistake is neglecting to consider both sides of the inequality when working with absolute values. Always ensure to split the absolute value inequality into two parts and solve them both.
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