Problem 40 - College Algebra with Corequisite Support

Question

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable \(x\) for the score.

Step-by-Step Solution

Answer

The inequality is \(|x - 82| \leq 18\).

Step 1: Identify the key components

Students need to score within 18 points of 82 to pass the test. This means the difference between a student's score, represented by the variable \(x\), and 82 must be 18 or less.

Step 2: Set up the absolute value inequality

The phrase 'within 18 points' translates to the absolute value inequality \(|x - 82| \leq 18\). This represents that the distance between \(x\) and 82 on the number line is at most 18.

Step 3: Short Answer

The inequality is \(|x - 82| \leq 18\).

Key Concepts

Inequalities

Inequalities are like the siblings of equations in algebra. While equations show that two values are the same, inequalities show how one value is different from another. Think of inequalities as a way to compare values, like saying one number is greater, lesser, or not more than another. In the context of the exercise, the inequality \(|x - 82| \leq 18\) indicates a range of scores for passing. Here, it highlights that the score, represented by the variable \(x\), must be no more than 18 points away from 82, either higher or lower.

It's super useful to know some common inequality symbols: \(<\) means "less than", \(>\) means "greater than", \(\leq\) means "less than or equal to", and \(\geq\) means "greater than or equal to". When you put inequalities into practice, you're working to find all possible numbers that make the inequality true, like a treasure hunt for all acceptable answers.

Distance on Number Line

The concept of distance on a number line is closely tied to absolute values. Imagine a straight path, where each step you take left or right represents a number. The absolute value of a number, such as \(|x - 82|\), signifies its distance from zero on this path. It’s like measuring with a ruler, where you only count the spaces, ignoring direction.

In our example, the goal is to express how close a student's score should be to the magic number 82 to pass the test. The inequality \(|x - 82| \leq 18\) tells us that the score is at most 18 steps away from 82. It ensures the distance between two points on the line (the ideal score and the student's score) doesn't exceed 18. This visualization is powerful as it helps you see where your solutions "live" on the number line.

Algebra Problems

Algebra problems often involve finding unknown values that satisfy certain conditions. In these problems, variables like \(x\) represent those unknowns. When tackling algebraic inequalities, you're essentially trying to crack the code of which values fit within the given parameters.

Solving the inequality \(|x - 82| \leq 18\) involves identifying all possible scores that fall within a safe range. To break it down: the inequality says the distance from 82 shouldn’t exceed 18. You split it into two simpler inequalities: \(x - 82 \leq 18\) and \(x - 82 \geq -18\). Solving these gives you the range \(64 \leq x \leq 100\). This means any score between 64 and 100 (inclusive) is acceptable.

• Step 1: Split the inequality to \(x - 82 \leq 18\) and \(x - 82 \geq -18\).
• Step 2: Solve for \(x\) in both cases.
• Step 3: Combine to find the range.

Such problems sharpen your problem-solving skills, as they require both understanding and application of algebraic concepts.