Question

The heights \(h\) of two-thirds of a population satisfy the inequality \(|h-68.5| \leq 2.7\) where \(h\) is measured in inches. Determine the interval on the real number line in which these heights lie.

Short answer

The interval on the real number line in which these heights lie is [65.8, 71.2] inches.

Step-by-step solution

Interpret the absolute value inequality

An absolute value inequality like \(|h-68.5| \leq 2.7\) can be rewritten as two distinct inequalities connected by 'and'. It translates to \(-2.7 \leq h-68.5 \leq 2.7\). This form of the inequality represents a range of values located around average height 68.5 inches, from -2.7 inches to 2.7 inches of deviation.

Solve both parts of the inequality

Next, separate the middle inequality into two parts: \(-2.7 \leq h-68.5\) and \(h-68.5 \leq 2.7\), and solve each by adding 68.5 to each side. This gives an interval for \(h\) from 65.8 inches to 71.2 inches.

Express the answer as an interval

The heights \(h\) lies between 65.8 and 71.2 inches. Therefore, the resulting interval for possible values of \(h\) can be expressed in interval notation as [65.8, 71.2] which represents every number between and including 65.8 inches and 71.2 inches.

Key concepts

Inequality Solving

In the fascinating world of algebra, inequality solving is akin to unlocking the secrets of a numerical treasure chest. It's about finding the range of possible values for a variable that make the inequality true. Imagine a seesaw where your goal is to maintain balance not at one specific point, but across a stretch of possibilities.

To tackle inequalities, especially those donned in the intricate armor of absolute values, one must embrace a two-pronged approach. Just as you learned in the exercise with heights, an absolute value inequality such as \(|h-68.5| \leq 2.7\) conceals two individual inequalities. These are the hidden gems: \(h-68.5 \leq 2.7\) and the less obvious counterpart, \(-2.7 \leq h-68.5\), each leading to a distinct set of solutions that, in the end, reveal the permissible range for our variable, \(h\).

Approaching each part separately simplifies the process. Add or subtract numbers strategically, similar to peeling off the layers of an onion, and the true essence of the inequality will be brought to light. With practice, these fundamental steps help you unveil the range of values that form the solution without losing sight of the elegance and balance algebra demands.

Real Number Line Intervals

Visualizing solutions to inequalities often leads us down a path sprinkled with dashes and dots, known as the real number line. This line serves as a map to navigate the vast ocean of numbers, pointing towards the specific regions where our solutions reside.

In the given exercise, the interval marking the heights of the population falls comfortably between two points: 65.8 and 71.2 inches. When plotting these on the real number line, you create a closed segment indicating that every tiny point on the line between these 'boundary markers' is part of the solution set.

How does this help?

By 'seeing' numbers in their natural habitat, your intuition for number relationships strengthens. Crucially, it aids your understanding of how inequalities partition the real number line into segments or rays, which represent a multitude of potential solutions.

Interval Notation

Conveying the breadth of solutions to an inequality in a crisp, concise fashion is the realm of interval notation. This mathematical shorthand captures the essence of a solution set without the fanfare of detailed explanation.

The solution set for our height inequality, [65.8, 71.2], is an example of interval notation in its prime. The square brackets signal that 65.8 and 71.2 are included in the set—they're part of the club. This precise, minimalistic notation informs us loudly and clearly that every real number between those two is invited to the party as well.

And just like that, with the power of interval notation, you've neatly packaged an infinite set of heights into a compact, easy-to-understand format. It's a highly efficient means of communicating complex mathematical concepts without overpowering the uninitiated, allowing even those early in their algebraic journey to grasp the vastness of solutions with simplicity and clarity.