Problem 31 Solve each inequality. Graph the... [FREE SOLUTION]

Chapter 1: Problem 31

Solve each inequality. Graph the solution set on a number line. $$ \frac{|n-3|}{2} < n $$

Short Answer

Expert verified

The solution set is $ n > 1 $, graph with an open circle at $ n = 1 $ and a line extending right.

Step by step solution

Understand the Inequality

The inequality to solve is $ \frac{|n-3|}{2} < n $. This involves an absolute value, which will require us to consider two cases: one for when the expression inside the absolute value is non-negative and one for when it is negative.

Split into Cases

Since the inequality involves absolute value $ |n-3| $, we split it into two cases. Case 1: $ n - 3 \geq 0 $ or $ n \geq 3 $. In this case, $ |n-3| = n-3 $. Case 2: $ n - 3 < 0 $ or $ n < 3 $. In this case, $ |n-3| = -(n-3) $.

Solve Case 1

In Case 1 ($ n \geq 3 $), substitute $ |n-3| = n-3 $ into the inequality: $ \frac{n-3}{2} < n $. Multiply through by 2 to get: $ n-3 < 2n $. Rearrange to find: $ n > 3 $. Since $ n \geq 3 $, combine with $ n > 3 $ to get $ n > 3 $.

Solve Case 2

In Case 2 ($ n < 3 $), substitute $ |n-3| = -(n-3) $ into the inequality: $ \frac{-(n-3)}{2} < n $. Simplify to $ \frac{-n + 3}{2} < n $, which becomes $ 3 < 3n $ after multiplying through by 2. Divide both sides by 3 to get $ 1 < n $ or $ n > 1 $. Since $ n < 3 $, combine the results to get $ 1 < n < 3 $.

Combine Results

Combine the solutions from both cases. From Case 1, we found $ n > 3 $. From Case 2, we found $ 1 < n < 3 $. Therefore, the solution set is $ n > 1 $, which includes both intervals.

Graph the Solution Set

On a number line, draw an open circle at $ n = 1 $ and extend the line to the right to indicate $ n > 1 $. This means that all numbers greater than 1 satisfy the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. Think of it as stripping away the sign to focus purely on magnitude. For an expression like $|x|$, it simply means "how far is x from zero without considering if x is positive or negative?" In our exercise, the expression $|n-3|$ asks for the distance of $n$ from 3.

If $n - 3 \geq 0$, then $|n-3| = n-3$ because the difference is non-negative and needs no adjustment.
If $n - 3 < 0$, then $|n-3| = -(n-3)$ because the difference is negative and we use a negative sign to convert it to positive distance.

This dual-case approach helps resolve any absolute value expressions, providing clarity regardless of the number's position relative to the reference point.

Number Line Graph

Graphing the solution to an inequality requires showing visually which values satisfy the condition. The number line is a simple yet effective tool for this. It visually represents numbers, with smaller numbers to the left and larger numbers to the right.
For our solution, the resulting inequality $n > 1$ is graphed on the number line by:

Drawing an open circle at $n = 1$ to indicate that 1 is not part of the solution set.
Extending a line to the right from this point, showing that all numbers greater than 1 satisfy the inequality.

Open circles indicate values not included, while filled circles would indicate inclusivity. This simple graph conveys complex algebraic solutions effectively at a glance.

Solution Set

A solution set in mathematics denotes all values that satisfy a given inequality or equation. For the original problem, after applying case logic due to the presence of the absolute value function, we derive a solution set of $n > 1$.
In our context:

From Case 1, valid values were $n > 3$.
From Case 2, we identified the range $1 < n < 3$.

Combining these, we recognize that any $n > 1$ satisfies the inequality, which includes both of these intervals. Understanding solution sets informs us of all potential values for the variable that make the original inequality true.

Algebraic Manipulation

Algebraic manipulation involves rearranging parts of an equation or inequality to achieve a clearer form or solution. This is a central part of solving inequalities and equations.
For our inequality $ \frac{|n-3|}{2} < n \,$ we need manipulation steps like:

Clearing fractions by multiplying through by the denominator (in this case, 2). This helps simplify the comparison without altering the solution.
Isolating terms. For example, $n-3 < 2n$ in Case 1 turns into $1 < n$ by combining like terms and simplifying the expression.

Such steps illuminate pathways to solutions in algebraic problems, simplifying complex expressions into more manageable forms, and ultimately leading to our final solution.

91影视

Solve each inequality. Graph the solution set on a number line. $$ \frac{|n-3|}{2} < n $$

Short Answer

Step by step solution

Understand the Inequality

Split into Cases

Solve Case 1

Solve Case 2

Combine Results

Graph the Solution Set

Key Concepts

Absolute Value

Number Line Graph

Solution Set

Algebraic Manipulation

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