Problem 110 Solve for x and y. Assume that a... [FREE SOLUTION]

91影视

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 110

Solve for x and y. Assume that a, b, c, d, and m are positive constants. $$ c(6 x-4)d $$

Short Answer

Expert verified

x < \frac{3d + 4c}{6c - 2d}

Step by step solution

Expand and Distribute the Constants

Distribute the constants on both sides of the inequality. This gives: \[ c(6x - 4) < d(3 + 2x) \]Expand to: \[ 6cx - 4c < 3d + 2dx \]

Move Terms Involving x to One Side

To isolate terms with $x$, move all $x$ terms to one side of the inequality: \[ 6cx - 2dx < 3d + 4c \]

Factor Out x

Factor out $x$: \[ x(6c - 2d) < 3d + 4c \]

Solve for x

Solve for $x$ by dividing both sides by $6c - 2d$ and making sure the direction of the inequality sign remains correct (note: $6c - 2d$ is positive because we are given $3c > d$): \[ x < \frac{3d + 4c}{6c - 2d} \]

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Key Concepts

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

Inequality Expansion

Inequality expansion involves 'opening up' expressions that are multiplied by a common term. In our problem, we start with: c(6x - 4) < d(3 + 2x). Using the distributive property, we expand each side: 6cx - 4c < 3d + 2dx. This step is crucial because it simplifies the inequality, making it easier to isolate variables and solve the problem.

Distributive Property

The distributive property states that multiplying a sum by a number is the same as doing each multiplication separately. For example, in our inequality: c(6x - 4) < d(3 + 2x), we distribute c and d across their respective parentheses: 6cx - 4c < 3d + 2dx. This operation breaks down complex expressions, simplifying them into manageable parts.

Variable Isolation

To solve inequalities, isolating the variable is a key step. After expanding, we reorganize the equation to gather all terms with the variable on one side: 6cx - 4c < 3d + 2dx becomes: 6cx - 2dx < 3d + 4c. This rearrangement helps prepare the inequality for easier solving. Remember, moving terms across the inequality sign requires reversing their signs.

Factoring Terms

Factoring is a technique used to simplify expressions. Once we have all like terms on one side of the inequality, we can factor out the common variable: 6cx - 2dx < 3d + 4c becomes: x(6c - 2d) < 3d + 4c. This step makes it easy to isolate x. Finally, we solve for x by dividing both sides by the coefficient of x, ensuring the inequality direction stays the same as long as we divide by a positive number.

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91影视

Solve for x and y. Assume that a, b, c, d, and m are positive constants. $$ c(6 x-4)d $$

Short Answer

Step by step solution

Expand and Distribute the Constants

Move Terms Involving x to One Side

Factor Out x

Solve for x

Key Concepts

Inequality Expansion

Distributive Property

Variable Isolation

Factoring Terms

One App. One Place for Learning.

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