/*! 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 8 Solve each equation. $$ \sqr... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 8

Solve each equation. $$ \sqrt{x+1}=7 $$

Short Answer

Expert verified
x = 48

Step by step solution

01

Isolate the Square Root

The equation is already isolated with the square root on one side: \[ \sqrt{x+1} = 7 \]
02

Square Both Sides

Remove the square root by squaring both sides of the equation: \[ (\sqrt{x+1})^2 = 7^2 \] \[ x+1 = 49 \]
03

Solve for x

Subtract 1 from both sides to solve for \(x\): \[ x + 1 - 1 = 49 - 1 \] \[ x = 48 \]

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Key Concepts

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

Isolating the Variable
When solving square root equations, the first step is to isolate the variable inside the square root. In our example, the equation we start with is already isolated: \ \( \sqrt{x+1} = 7 \).\
By isolating the variable, we ensure there's a clear path to remove the square root. Always aim to have the square root term by itself on one side of the equation.
Tips to isolate variables include:
  • Combining like terms
  • Moving constants to the other side of the equation
  • Using inverse operations (like addition or subtraction)
By doing this, it'll make the subsequent steps simpler and more straightforward.
Squaring Both Sides
After isolating the variable inside the square root, the next step is to remove the square root by squaring both sides of the equation. In the example: \ \( (\sqrt{x+1})^2 = 7^2 \)\ Squaring the term \( \sqrt{x+1} \) will get rid of the square root, because the square of a square root returns the original value inside the root. It simplifies the equation to \( x+1 = 49 \).

Be mindful when squaring both sides:
  • Square the entire sides of the equation, not just part of it.
  • Remember to maintain equality; both sides must be squared.
Squaring both sides is a crucial step that transforms a square root equation into a simpler linear equation.
Algebraic Manipulation
The final step involves solving the simplified equation using algebraic techniques. After squaring, we had: \ \( x+1 = 49 \)\ Now, we isolate \( x \) by subtracting 1 from both sides: \ \( x + 1 - 1 = 49 - 1 \) So, \ \( x = 48 \).

A few tips for algebraic manipulation:
  • Perform the same operation on both sides of the equation to maintain balance.
  • Combine like terms and simplify where possible.
  • Always double-check your steps.
These techniques ensure you clearly work towards the solution, maintaining accuracy in your calculations.

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Most popular questions from this chapter

Every real number is a complex number. Explain why this is so.

Perform the indicated operations. Give answers in standard form. $$ \left(\frac{2+i}{2-i}+\frac{i}{1+i}\right) i $$

Find each quotient. $$ \frac{-38-8 i}{7+3 i} $$

Concept Check Let \(a=1\) and let \(b=64\) (a) Evaluate \(\sqrt{a}+\sqrt{b} .\) Then find \(\sqrt{a+b} .\) Are they equal? (b) Evaluate \(\sqrt[3]{a}+\sqrt[3]{b}\). Then find \(\sqrt[3]{a+b} .\) Are they equal? (c) Complete the following: In general, \(\sqrt[n]{a}+\sqrt[n]{b} \neq\) ________ based on the observations in parts (a) and (b) of this exercise.

Write each number as a product of a real number and i. Simplify all radical expressions. $$ \sqrt{-96} $$
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