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Chapter 8: Problem 68

Graph and write interval notation for each compound inequality. $$ t>4 \text { or } t>-1 $$

Short Answer

Expert verified
The interval notation is \((-1, \infty)\).

Step by step solution

01

Understand the Compound Inequality

A compound inequality involving 'or' means that either one of the inequalities or both can be true. In this case, the inequalities are:1. \(t > 4\)2. \(t > -1\)
02

Solve Each Inequality Separately

Solve the inequalities individually:1. \(t > 4\) means any value of \(t\) that is greater than 4.2. \(t > -1\) means any value of \(t\) that is greater than -1.
03

Combine the Solutions

Since the compound inequality uses 'or,' we'll combine the solutions. The solution set includes any value that satisfies either \(t > 4\) or \(t > -1\). Since \(t > -1\) already encompasses the \(t > 4\) values, we take the more inclusive inequality: \(t > -1\).
04

Write the Interval Notation

The interval notation for \(t > -1\) is \((-1, \infty)\).
05

Graph the Solution

To graph \(t > -1\), draw a number line, mark a point at \(-1\) with an open circle (since \-1\ is not included), and shade the line to the right of \(-1\) extending to infinity.

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Key Concepts

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

graphing inequalities
When we talk about graphing inequalities, we're visually representing the solution sets on a number line. For the inequality \( t > -1 \), you start by drawing a number line. Find the point corresponding to \( -1 \). Since \( t > -1 \) excludes \( -1 \), we use an open circle to indicate that \( -1 \) isn't part of the solution. Next, we shade the line to the right of \( -1 \) to show all numbers greater than \( -1 \). This shaded area represents all possible values of \( t \) that satisfy the inequality.
interval notation
Interval notation is a method to express the set of solutions in a compact form. For inequalities like \( t > -1 \), the interval notation helps to convey this idea efficiently. We use parentheses for non-inclusive boundaries. So, for \( t > -1 \), we write the interval notation as \( (-1, \infty) \). The open parenthesis at \( -1 \) means \( -1 \) is not included in the set. The infinity symbol \( \infty \) indicates the set extends indefinitely. Remember, infinity is always written with a parenthesis because it's not a number but an idea of boundlessness.
inequality solutions
Solving inequalities involves finding all values that satisfy the given condition. For compound inequalities, we consider whether the inequalities are connected by 'and' or 'or'. An 'or' compound inequality, like \( t > 4 \) or \( t > -1 \), is true if at least one part of the inequality is true. Individually, \( t > 4 \) includes all values greater than 4, and \( t > -1 \) includes all values greater than -1. Since \( t > -1 \) already includes all values where \( t > 4 \), we choose the more inclusive inequality: \( t > -1 \). This solution set is then written in interval notation as \( (-1, \infty) \) and graphed accordingly to show all valid solutions.

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Most popular questions from this chapter

To prepare for Chapter \(9,\) review solving systems of equations using elimination (Section 4.3). Solve. [ 4.3] $$ \begin{aligned} &4 x-1=3 y\\\ &2 y-4=5 x \end{aligned} $$

Solve. $$ (x-1)(x+2)(x-4) \geq 0 $$

On many graphing calculators, the TEST key provides access to inequality symbols, while the LOGIC option of that same key accesses the conjunction and and the disjunction or. Thus, if \(y_{1}=x>-2\) and \(y_{2}=x<4,\) Exercise 55 can be checked by forming the expression \(y_{3}=y_{1}\) and \(y_{2} .\) The interval(s) in the solution set appears as a horizontal line 1 unit above the \(x\) -axis. (Be careful to "deselect" \(y_{1}\) and \(y_{2}\) so that only \(y_{3}\) is drawn.) Use the Test key to check Exercises \(59,63,65,\) and 67

For \(f(x)=\frac{5-2 x}{4 x+3},\) find all \(x\) -values for which \(f(x) \geq 0\).

Solve. $$ t-2 \leq|t-3| $$
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