Ch. 1 - Equations and Inequalities

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

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 48a

Chapter 2, Problem 48a

Write each English sentence as an equation in two variables. Then graph the equation. The y-value is the difference between four and twice the x-value.

Verified step by step guidance

Step 1: Start by translating the English sentence into a mathematical equation. The sentence states that the y-value is the difference between four and twice the x-value. This can be written as:

y = 4 - 2x

Step 2: Identify the two variables in the equation. Here,

x

is the independent variable, and

y

is the dependent variable.

Step 3: To graph the equation, create a table of values. Choose a few values for

x

(e.g., -2, -1, 0, 1, 2) and calculate the corresponding

y

values using the equation

y = 4 - 2x

Step 4: Plot the points from the table of values on a coordinate plane. For example, if

x = 0

, then

y = 4

, so one point is (0, 4). Repeat this for all chosen

x

values.

Step 5: Draw a straight line through the plotted points, as the equation

y = 4 - 2x

represents a linear relationship. Label the graph appropriately with axes and scales.

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Key Concepts

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

Linear Equations

Linear equations are mathematical statements that express a relationship between two variables, typically in the form y = mx + b, where m is the slope and b is the y-intercept. In this context, the equation represents a straight line on a graph, which can be derived from the given English sentence.

Graphing Equations

Graphing equations involves plotting points on a coordinate plane to visually represent the relationship between the variables. For linear equations, this results in a straight line, which can be created by identifying key points, such as the y-intercept and additional points derived from the equation.

Translating English Sentences to Equations

Translating English sentences into mathematical equations requires understanding the relationships described in the sentence. In this case, phrases like 'the difference between' and 'twice the x-value' indicate how to structure the equation, leading to a mathematical representation of the verbal statement.

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Related Practice

Textbook Question

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 4(3x - 2) - 3x < 3(1 + 3x) - 7

Textbook Question

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 1/(x - 1) + 5 = 11/(x - 1)

Textbook Question

Solve each equation in Exercises 41–60 by making an appropriate substitution. x^(-2) - x^(-1) - 6 = 0

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Textbook Question

Write each English sentence as an equation in two variables. Then graph the equation. The y-value is four more than twice the x-value.

Textbook Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} - 2 x = 2$

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