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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

01

Expand and Distribute the Constants

Distribute the constants on both sides of the inequality. This gives: \[ c(6x - 4) < d(3 + 2x) \]Expand to: \[ 6cx - 4c < 3d + 2dx \]
02

Move Terms Involving x to One Side

To isolate terms with \(x\), move all \(x\) terms to one side of the inequality: \[ 6cx - 2dx < 3d + 4c \]
03

Factor Out x

Factor out \(x\): \[ x(6c - 2d) < 3d + 4c \]
04

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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Most popular questions from this chapter

College Faculty. The number of part-time instructional faculty in U.S. postsecondary institutions is growing at a greater rate than the number of full-time faculty. The number of parttime faculty, in thousands, is approximated by $$ p(t)=27 t+325 $$ and the number of full-time faculty, in thousands, is approximated by $$ f(t)=16 t+500 $$ For both functions, \(t\) represents the number of years after \(1995 .\) Using an inequality, determine those years for which there were more part-time faculty than full-time faculty.

For \(f(x)\) as given, use interval notation to write the domain of \(f\). $$ f(x)=\frac{\sqrt{3-4 x}}{x+7} $$

Write an equivalent inequality using absolute value. $$ -5

Solve. $$ \begin{aligned} &4 x-y=1\\\ &y=7-x \end{aligned} $$

Simplify. $$ -3.85+4.2 $$
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