/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 75 For the following problems, solv... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 75

For the following problems, solve the equations. $$ \sqrt{2 x+3}+8=11 $$

Short Answer

Expert verified
Short answer: Following the steps of isolating the square root term, squaring both sides of the equation, simplifying the equation, solving for x, and checking for extraneous solutions, we find that the solution to the given equation is x = 3.

Step by step solution

01

Isolate the square root

First, isolate the square root term by subtracting 8 from both sides of the equation: $$ \sqrt{2x + 3} + 8 - 8 = 11 - 8 $$ Now, the equation looks like: $$ \sqrt{2x + 3} = 3 $$
02

Square both sides of the equation

To eliminate the square root, square both sides of the equation: $$ (\sqrt{2x + 3})^2 = 3^2 $$ This simplifies to: $$ 2x + 3 = 9 $$
03

Simplify the equation and solve for x

Now, solve for x by subtracting 3 from both sides of the equation: $$ 2x + 3 - 3 = 9 - 3 $$ This gives us: $$ 2x = 6 $$ Next, divide both sides by 2 to find the value of x: $$ \frac{2x}{2} = \frac{6}{2} $$ So, the solution is: $$ x = 3 $$
04

Check the solution

Finally, check if the solution x = 3 is not an extraneous solution by plugging it back into the original equation: $$ \sqrt{2(3) + 3} + 8 = 11 $$ Simplify the terms under the square root: $$ \sqrt{6 + 3} + 8 = 11 $$ This gives us: $$ \sqrt{9} + 8 = 11 $$ Since: $$ 3 + 8 = 11 $$ The solution x = 3 is valid, and there are no extraneous solutions. So the answer is: $$ x = 3 $$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Square Roots
Square roots are mathematical functions that help us determine what number, when multiplied by itself, equals a given input value. When solving equations that contain square roots, it is crucial to understand that they have both a positive and a negative root. However, in many real-world applications, particularly those involving lengths or distances, we often use only the non-negative root.
In a problem, when you see a square root, it might be tempting to immediately square both sides, but remember that it's often best to first isolate the square root term. This simplification step can make solving the equation much easier, reducing the complexity of your calculations.
Isolating Terms
Isolating terms in an equation generally means performing algebraic manipulations to get the term we want to solve for by itself on one side of the equation. This is a fundamental step when solving equations, especially when dealing with variables.
For example, consider the equation \(\sqrt{2x + 3} + 8 = 11\). Our target is to isolate \(\sqrt{2x + 3}\), achieved by subtracting 8 from both sides. This action simplifies the equation, paving the way to address the square root directly and solve for the variable effectively.
Extraneous Solutions
Extraneous solutions are solutions derived from the manipulation of an equation that do not satisfy the original equation. These often arise in problems involving squaring both sides of an equation, as this can potentially introduce false roots.
After solving an equation, especially those involving square roots or other radicals, always substitute the solutions back into the original equation to verify their correctness. This helps ensure that no extraneous solutions were introduced during the algebraic process, guaranteeing the validity of your solution.
Algebraic Manipulation
Algebraic manipulation involves using arithmetic operations and algebraic processes to transform equations into a more solvable form. This is a critical skill in problem-solving and can involve various techniques such as isolating terms, distributing, and combining like terms.
In the given exercise, algebraic manipulation is used multiple times. Initially, to isolate the square root term, then later to simplify the resulting equation by solving for the variable. Familiarity with these techniques is essential for effectively handling more complex equations in algebra.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Simplify each expression by performing the indicated operation. $$ (\sqrt{12}+5 \sqrt{3})(2 \sqrt{3}-2 \sqrt{12}) $$

Simplify each expression by performing the indicated operation. $$ \frac{1+\sqrt{6}}{1-\sqrt{6}} $$

Simplify each expression by performing the indicated operation. $$ (\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2}) $$

For the following problems, simplify each of the square root expressions. $$ (4-\sqrt{5 y})^{2} $$

For the following problems, solve the equations. $$ \sqrt{4 a}=16 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.