Answer Using an inequality notation or inequality interval to express the solution of an inequality depends on the nature of the solution. If there are infinite number of values of given variable in the inequality, that satisfy the given inequality, it’s always better to express the inequality notion like > , < , ≥ , ≤ as it shows an infinite range of solutions. Example: 1 − 2 x ≤ 3 1 − 2 x − 1 ≤ 3 − 1 − 2 x ≤ 2 − 2 x − 2 ≥ 2 − 2 x ≥ − 1 The solution set is { x | x ≥ − 1 } If there are finite number of values of given variable in the inequality, that satisfy the given inequality, it’s always better to express the inequality interval. Example: 4 ≤ 2 x + 2 ≤ 10 4 − 2 ≤ 2 x + 2 − 2 ≤ 10 − 2 2 ≤ 2 x ≤ 8 2 2 ≤ 2 x 2 ≤ 8 2 1 ≤ x ≤ 4 The solution set is { x | 1 ≤ x ≤ 4 } or using interval notation, all numbers in the interval [ 1 , 4 ] Explanation Using an inequality notation or inequality interval to express the solution of an inequality depends on the nature of the solution. If there are infinite number of values of given variable in the inequality, that satisfy the given inequality, it’s always better to express the inequality notion like > , < , ≥ , ≤ as it shows an infinite range of solutions. Example: 1 − 2 x ≤ 3 1 − 2 x − 1 ≤ 3 − 1 − 2 x ≤ 2 − 2 x − 2 ≥ 2 − 2 x ≥ − 1 The solution set is { x | x ≥ − 1 } If there are finite number of values of given variable in the inequality, that satisfy the given inequality, it’s always better to express the inequality interval. Example: 4 ≤ 2 x + 2 ≤ 10 4 − 2 ≤ 2 x + 2 − 2 ≤ 10 − 2 2 ≤ 2 x ≤ 8 2 2 ≤ 2 x 2 ≤ 8 2 1 ≤ x ≤ 4 The solution set is { x | 1 ≤ x ≤ 4 } or using interval notation, all numbers in the interval [ 1 , 4 ]

Precalculus Enhanced with Graphing Utilities

6th Edition

ISBN: 9780321795465

Author: Michael Sullivan, Michael III Sullivan

Publisher: PEARSON

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Question

Chapter A.9, Problem 130AYU

To determine

To find: Do you prefer to use inequality notation or interval notation to express the solution to an inequality? Give your reasons. Are there particular circumstances when you prefer one to the other? Cite examples.

Expert Solution

Answer to Problem 130AYU

Using an inequality notation or inequality interval to express the solution of an inequality depends on the nature of the solution.

If there are infinite number of values of given variable in the inequality, that satisfy the given inequality, it’s always better to express the inequality notion like $>, <, \geq, \leq$ as it shows an infinite range of solutions.

Example:

$\begin{array}{l} 1 - 2 x \leq 3 \\ 1 - 2 x - 1 \leq 3 - 1 \\ - 2 x \leq 2 \\ \frac{- 2 x}{- 2} \geq \frac{2}{- 2} \\ x \geq - 1 \end{array}$

The solution set is ${x | x \geq - 1}$

If there are finite number of values of given variable in the inequality, that satisfy the given inequality, it’s always better to express the inequality interval.

Example:

$\begin{array}{l} 4 \leq 2 x + 2 \leq 10 \\ 4 - 2 \leq 2 x + 2 - 2 \leq 10 - 2 \\ 2 \leq 2 x \leq 8 \\ \frac{2}{2} \leq \frac{2 x}{2} \leq \frac{8}{2} \\ 1 \leq x \leq 4 \end{array}$

The solution set is ${x | 1 \leq x \leq 4}$ or using interval notation, all numbers in the interval $[1, 4]$

Explanation of Solution

Using an inequality notation or inequality interval to express the solution of an inequality depends on the nature of the solution.

Example:

$\begin{array}{l} 1 - 2 x \leq 3 \\ 1 - 2 x - 1 \leq 3 - 1 \\ - 2 x \leq 2 \\ \frac{- 2 x}{- 2} \geq \frac{2}{- 2} \\ x \geq - 1 \end{array}$

The solution set is ${x | x \geq - 1}$

If there are finite number of values of given variable in the inequality, that satisfy the given inequality, it’s always better to express the inequality interval.

Example:

$\begin{array}{l} 4 \leq 2 x + 2 \leq 10 \\ 4 - 2 \leq 2 x + 2 - 2 \leq 10 - 2 \\ 2 \leq 2 x \leq 8 \\ \frac{2}{2} \leq \frac{2 x}{2} \leq \frac{8}{2} \\ 1 \leq x \leq 4 \end{array}$