Problem 87 - A Graphical Approach to College Algebra

Question

Solve each problem. Dr. Cazayoux has found that, over the years, \(95 \%\) of the babies he delivered weighed \(x\) pounds, where \(|x-8.0| \leq 1.5 .\) What range of weights corresponds to this inequality?

Step-by-Step Solution

Answer

The range of weights is 6.5 to 9.5 pounds.

Step 1: Understand the Inequality

The inequality given is \(|x-8.0| \leq 1.5\). This inequality represents the range of values that \(x\) can take, where \(x\) is the weight of the babies in pounds.

Step 2: Break Down the Absolute Value

The inequality \(|x-8.0| \leq 1.5\) can be broken into two separate inequalities: \(x - 8.0 \leq 1.5\) and \(x - 8.0 \geq -1.5\). This is how absolute values work, representing the distance from zero.

Step 3: Solve the First Inequality

Take the inequality \(x - 8.0 \leq 1.5\) and solve for \(x\). Add 8.0 to both sides to find:\[x \leq 9.5\]

Step 4: Solve the Second Inequality

Now solve the inequality \(x - 8.0 \geq -1.5\). Add 8.0 to both sides to find:\[x \geq 6.5\]

Step 5: Combine the Solutions

Combine the solutions of the two inequalities to express the range of values for \(x\). The combined solution is:\[6.5 \leq x \leq 9.5\]

Step 6: Interpret the Result

The combined inequality \(6.5 \leq x \leq 9.5\) means that the weights of the babies are between 6.5 pounds and 9.5 pounds inclusive.

Key Concepts

Absolute Value

Absolute value is a way to measure the distance a number is from zero on the number line, without considering the direction. It's always a positive number or zero. In this exercise, the absolute value inequality is \(|x - 8.0| \leq 1.5\). This tells us how far the weights \(x\) can be from 8 pounds. It shows that the weight could be at most 1.5 pounds more or less than 8.
This concept of absolute value helps to simplify and solve inequalities by stripping negative signs. It translates to two inequalities: one for the positive distance and one for the negative, aiding in identifying the complete range of possible values.

Range of Values

The range of values in inequalities indicates the span of all possible numbers that satisfy the condition. For \(|x - 8.0| \leq 1.5\), we translate this into two parts: \(x - 8.0 \leq 1.5\) and \(x - 8.0 \geq -1.5\).

• The first inequality \(x - 8.0 \leq 1.5\) simplifies to \(x \leq 9.5\).
• The second one \(x - 8.0 \geq -1.5\) simplifies to \(x \geq 6.5\).

By combining these results, we find the full range where \(x\) can fall: \[6.5 \leq x \leq 9.5\]
This complete range tells us that the baby weights are between 6.5 and 9.5 pounds, inclusive. Understanding range allows us to interpret and apply inequalities in real-world scenarios effectively.

Solving Inequalities

Solving inequalities involves finding all values of a variable that make the inequality true. For absolute values, this process requires breaking them down into simpler expressions.
In our example, by breaking \(|x - 8.0| \leq 1.5\) into two inequalities, we treated them independently.

• For \(x - 8.0 \leq 1.5\), adding 8.0 gave \(x \leq 9.5\).
• For \(x - 8.0 \geq -1.5\), adding 8.0 gave \(x \geq 6.5\).

Each inequality was solved using basic algebra, combining simple addition to isolate \(x\). Finally, combining these inequalities into a single range \(6.5 \leq x \leq 9.5\) provides a clear solution.
These techniques are essential when working with any inequalities, helping to capture all possible solutions.