A number is less than 20 units away from 7 on the number line. What numbers are within these parameters? Choose the inequality that represents the situation. Solve the inequality.
Answer (2)
The absolute value inequality representing the situation is ∣ x − 7∣ < 20 . Solving this yields the compound inequality − 13 < x < 27 , meaning x is any number between -13 and 27. Thus, all numbers greater than -13 and less than 27 are within the specified range.
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Represent the problem as an absolute value inequality: ∣ x − 7∣ < 20 .
Rewrite the absolute value inequality as a compound inequality: − 20 < x − 7 < 20 .
Isolate x by adding 7 to all parts of the inequality: − 13 < x < 27 .
The solution is all numbers between -13 and 27: − 13 < x < 27 .
Explanation
Setting up the Inequality Let x be the number we are looking for. The problem states that this number is less than 20 units away from 7 on the number line. This means the distance between x and 7 is less than 20. We can write this as an absolute value inequality:
Expressing Distance as an Absolute Value The distance between x and 7 is represented by ∣ x − 7∣ . Since this distance is less than 20, we have the inequality:
The Absolute Value Inequality ∣ x − 7∣ < 20
Rewriting as a Compound Inequality To solve this absolute value inequality, we can rewrite it as a compound inequality:
The Compound Inequality − 20 < x − 7 < 20
Isolating x Now, we add 7 to all parts of the inequality to isolate x :
Adding 7 to All Sides − 20 + 7 < x − 7 + 7 < 20 + 7
Simplifying This simplifies to:
The Solution − 13 < x < 27
Interpreting the Solution This means that x is greater than -13 and less than 27. In other words, the numbers within these parameters are all the numbers between -13 and 27, not including -13 and 27.
Examples
Imagine you're setting up a game at a carnival where participants have to guess a number. You tell them the number is within 20 units of 7. This problem helps you determine the range of numbers they can guess. Understanding absolute value inequalities is useful in scenarios such as determining acceptable ranges in manufacturing, setting tolerance levels in engineering, or defining error margins in scientific experiments. This algebraic approach ensures precision and accuracy in practical tasks.
