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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 77a
Chapter 2, Problem 77a

List the quadrant or quadrants satisfying each condition. x3 > 0 and y3 <0

Verified step by step guidance
1
Understand the problem: We are tasked with identifying the quadrant(s) where the conditions x^3 > 0 and y^3 < 0 are satisfied. Recall that the Cartesian plane is divided into four quadrants, and the signs of x and y in each quadrant determine the conditions.
Analyze the condition x^3 > 0: The cube of x is positive when x itself is positive. This means x > 0. On the Cartesian plane, x > 0 corresponds to Quadrants I and IV.
Analyze the condition y^3 < 0: The cube of y is negative when y itself is negative. This means y < 0. On the Cartesian plane, y < 0 corresponds to Quadrants III and IV.
Combine the conditions: For both conditions to be true simultaneously, we need a quadrant where x > 0 and y < 0. From the analysis, this occurs in Quadrant IV.
Conclude the solution: The quadrant that satisfies the conditions x^3 > 0 and y^3 < 0 is Quadrant IV.

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Key Concepts

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

Quadrants of the Cartesian Plane

The Cartesian plane is divided into four quadrants based on the signs of the x and y coordinates. Quadrant I has both x and y positive, Quadrant II has x negative and y positive, Quadrant III has both x and y negative, and Quadrant IV has x positive and y negative. Understanding these quadrants is essential for determining where specific conditions hold true.
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Inequalities and Their Solutions

Inequalities express a relationship where one side is not equal to the other, often involving greater than (>) or less than (<) symbols. In this case, x^3 > 0 indicates that x must be positive, while y^3 < 0 indicates that y must be negative. Solving these inequalities helps identify the values of x and y that satisfy the given conditions.
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Cubic Functions and Their Behavior

Cubic functions, such as f(x) = x^3, exhibit specific behaviors based on the sign of x. For positive x, the output is positive, and for negative x, the output is negative. This characteristic is crucial for understanding the conditions x^3 > 0 and y^3 < 0, as it directly influences the signs of x and y in the context of the quadrants.
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Related Practice
Textbook Question

Compute the discriminant. Then determine the number and type of solutions for the given equation. x2 - 2x + 1 = 0

Textbook Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. |4x - 3| = |4x - 5|

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Textbook Question

In Exercises 59–94, solve each absolute value inequality. |3 - (2/3)x| > 5

Textbook Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. |3x - 1| = |x + 5|

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views
Textbook Question

In Exercises 59–94, solve each absolute value inequality. 3|x - 1| + 2 ≥ 8

Textbook Question

Solve each equation by the method of your choice. (x−3)2−25=0(x-3)^2 - 25 = 0