Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 109

Chapter 2, Problem 109

In Exercises 107–110, use graphs to find each set. [1,3) ∩ (0,4)

Verified step by step guidance

Step 1: Understand the notation. The interval [1,3) means all numbers from 1 to 3, including 1 but not including 3. The interval (0,4) means all numbers greater than 0 and less than 4, excluding both 0 and 4.

Step 2: Represent each interval on a number line. For [1,3), draw a solid dot at 1 (to indicate inclusion) and an open circle at 3 (to indicate exclusion). For (0,4), draw open circles at both 0 and 4.

Step 3: Identify the intersection ∩ of the two intervals. The intersection represents the set of numbers that are common to both intervals.

Step 4: On the number line, observe where the two intervals overlap. The overlap starts at 1 (included because of [1,3)) and ends at 3 (excluded because of [1,3)).

Step 5: Write the result of the intersection as an interval. The intersection is [1,3), which includes all numbers from 1 to 3, including 1 but not including 3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses brackets and parentheses to indicate whether endpoints are included or excluded. For example, [1, 3) means that 1 is included in the interval, while 3 is not, indicating all numbers from 1 up to but not including 3.

Set Intersection

Set intersection refers to the operation of finding common elements between two sets. The intersection of sets A and B, denoted as A ∩ B, includes only those elements that are present in both sets. In the context of intervals, this means identifying the overlapping range of values that satisfy both conditions.

Graphing Intervals

Graphing intervals involves visually representing the range of values on a number line. Each interval is depicted with a line segment, where closed intervals are marked with solid dots and open intervals with open dots. This visual representation helps in easily identifying overlaps and intersections between different intervals.

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Related Practice

Textbook Question

If a number is decreased by 3, the principal square root of this difference is 5 less than the number. Find the number(s).

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Textbook Question

In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = x² - 2x + 2

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Textbook Question

Solve and graph the solution set on a number line: (2x−3)/4 ≥ 3x/4 + 1/2

When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.