Domanda del libro di testo
In Exercises 99–106, solve each equation. 0.7x + 0.4(20) = 0.5(x + 20)
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 2, Problema 105
In Exercises 104–106, express each interval in set-builder notation and graph the interval on a number line. (-2, ∞)
Guida verificata passo dopo passo1
Understand the given interval: The interval (-2, ∞) represents all real numbers greater than -2 but not including -2 itself. The parentheses indicate that -2 is not part of the interval, and ∞ means the interval extends indefinitely in the positive direction.
Convert the interval to set-builder notation: In set-builder notation, we describe the set of numbers using a condition. For this interval, the condition is that x is greater than -2. This can be written as {x | x > -2}, where the vertical bar '|' means 'such that.'
Prepare to graph the interval on a number line: To graph the interval (-2, ∞), start by marking the point -2 on the number line. Since -2 is not included in the interval, use an open circle at -2 to indicate this.
Draw the graph: From the open circle at -2, draw a line extending to the right indefinitely to represent all numbers greater than -2. Add an arrow at the end of the line to show that the interval continues infinitely.
Review the solution: The interval (-2, ∞) is expressed in set-builder notation as {x | x > -2}, and the graph includes an open circle at -2 with a line extending to the right indefinitely.
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Set-Builder Notation
Set-builder notation is a mathematical shorthand used to describe a set by specifying a property that its members must satisfy. For example, the interval (-2, ∞) can be expressed in set-builder notation as {x | x > -2}, meaning 'the set of all x such that x is greater than -2'. This notation is particularly useful for defining intervals and sets in a concise manner.
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05:18
Interval Notation
Intervals
An interval is a range of numbers between two endpoints. Intervals can be open, closed, or half-open, depending on whether the endpoints are included. The interval (-2, ∞) is an open interval starting at -2 and extending indefinitely to the right, meaning -2 is not included in the set, but all numbers greater than -2 are.
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05:18Interval Notation
Graphing on a Number Line
Graphing an interval on a number line involves visually representing the range of values included in the interval. For the interval (-2, ∞), you would place an open circle at -2 to indicate that it is not included, and then shade the line to the right to show that all numbers greater than -2 are part of the interval. This visual representation helps in understanding the extent of the interval.
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02:35Graphing Lines in Slope-Intercept Form
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