Ch. 1 - Equations and Inequalities

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

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 40a

Chapter 2, Problem 40a

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 3x/5 - (x - 3)/2 = (x + 2)/3

Verified step by step guidance

Step 1: Identify the least common denominator (LCD) of all the denominators in the equation. The denominators are 5, 2, and 3. The LCD is 30.

Step 2: Multiply every term in the equation by the LCD (30) to eliminate the fractions. This means multiplying each term by 30 and simplifying.

Step 3: Distribute the multiplication across each term. For example, 30 * (3x/5) becomes (30 * 3x) / 5, which simplifies to 18x. Repeat this process for the other terms.

Step 4: After clearing the fractions, simplify the resulting equation by combining like terms and isolating the variable (x). This may involve distributing, combining terms, and moving terms across the equals sign.

Step 5: Solve for x by dividing or performing any necessary operations to isolate x completely. Check your solution by substituting it back into the original equation to ensure it satisfies the equation.

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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 the equality of two linear expressions. They typically take the form ax + b = c, where a, b, and c are constants, and x is the variable. Understanding how to manipulate these equations is essential for solving them, as it involves isolating the variable on one side of the equation.

Common Denominator

When dealing with fractions in linear equations, finding a common denominator is crucial for simplifying the equation. The common denominator allows you to eliminate the fractions by multiplying each term by this value, making it easier to solve for the variable. This step is particularly important when the equation contains multiple fractions with different denominators.

Isolating the Variable

Isolating the variable is a key step in solving linear equations, where the goal is to get the variable (e.g., x) alone on one side of the equation. This often involves performing inverse operations, such as addition, subtraction, multiplication, or division, to both sides of the equation. Mastery of this concept is essential for finding the solution to the equation.

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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. 4/x = 5/2x + 3

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Textbook Question

In Exercises 36–43, use the five-step strategy for solving word problems. An apartment complex has offered you a move-in special of 30% off the first month's rent. If you pay \$945 for the first month, what should you expect to pay for the second month when you must pay full price?

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Textbook Question

In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. x² + 3x

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Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 2 + √-4)²

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Textbook Question

In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? E = mc² for m

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Textbook Question

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. $(x^{2} - x - 4)^{\frac{3}{4}} - 2 = 6$

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