Question

Making Precision Ball Bearings The FireBall Company manufactures ball bearings for precision equipment. One of its products is a ball bearing with a stated radius of 3 centimeters \((\mathrm{cm}) .\) Only ball bearings with a radius within \(0.01 \mathrm{~cm}\) of this stated radius are acceptable. If \(x\) is the radius of a ball bearing, a formula describing this situation is $$ |x-3| \leq 0.01 $$

Short answer

The acceptable radius range for the ball bearings is 2.99 cm to 3.01 cm.

Step-by-step solution

Understand the inequality

The inequality \[ |x - 3| \leq 0.01 \] indicates that the difference between the radius of the ball bearing, \(x\), and the stated radius of 3 cm should be no more than 0.01 cm.

Rewrite the absolute value inequality

The absolute value inequality can be rewritten as two separate inequalities: \[ -0.01 \leq x - 3 \leq 0.01 \]

Solve the inequality for x

First, solve the left side of the inequality: \[ -0.01 \leq x - 3 \] Add 3 to both sides: \[ -0.01 + 3 \leq x \] Simplifying this gives: \[ 2.99 \leq x \] Next, solve the right side of the inequality: \[ x - 3 \leq 0.01 \] Add 3 to both sides: \[ x \leq 3 + 0.01 \] Simplifying this gives: \[ x \leq 3.01 \]

Combine the inequalities

Combine the two inequalities to express the range of acceptable values for x: \[ 2.99 \leq x \leq 3.01 \] This means the radius of acceptable ball bearings must be between 2.99 cm and 3.01 cm.

Key concepts

inequality solving

Understanding and solving inequalities is a key skill in algebra. Inequalities show the relationship between values that are not necessarily equal but fall within a certain range.
When we solve an inequality, we are finding all possible values of the variable that make the inequality true.
Unlike equations which have specific solutions, inequalities often have a range of solutions. For example, the inequality \( |x - 3| \leq 0.01 \) represents all the values of \( x \) that are within 0.01 units of 3 on the number line.
To solve absolute value inequalities, we break them down into two separate inequalities without the absolute value. This gives us the range in which the possible solutions lie.

absolute value

Absolute value represents the distance of a number from zero, regardless of direction. It is always non-negative.
For example, the absolute value of both 3 and -3 is 3, because both are 3 units away from zero on the number line.
In algebra, absolute value can be denoted as |x|, which represents the distance of x from zero.
When dealing with absolute value inequalities, such as \( |x - 3| \leq 0.01 \), it tells us the distance between x and 3 should not exceed 0.01.
We solve it by rewriting as two separate inequalities: \(-0.01 \leq x - 3 \leq 0.01\). This approach helps us find the range of values that satisfy the condition.

manufacturing tolerance

Manufacturing tolerance is about ensuring products meet specific dimensional limits. This ensures they are functional and compatible with other parts.
In this context, the FireBall Company makes ball bearings with a radius of 3 cm. However, imperfections can occur, so a tolerance of \(\text{0.01 cm} \) is permitted.
This tolerance indicates that any ball bearing with a radius between 2.99 cm and 3.01 cm is acceptable. Anything outside this range is considered defective.
Tolerance ensures quality control and reliability of manufactured products, critical in precision engineering industries.
The formula \( |x - 3| \leq 0.01 \) captures this tolerance mathematically, ensuring only ball bearings within the permitted deviation are used.