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Chapter 3: Problem 11

Solve each equation. If it has no solution, write "no solution." $$ \sqrt{x-3}=9 $$

Short Answer

Expert verified
The solution is \( x = 84 \).

Step by step solution

01

Isolate the Square Root

The equation is already in the form where the square root is isolated: \( \sqrt{x-3} = 9 \).
02

Square Both Sides

To eliminate the square root, square both sides of the equation: \( (\sqrt{x-3})^2 = 9^2 \). This simplifies to: \( x-3 = 81 \).
03

Solve for x

Add 3 to both sides to isolate \( x \): \( x-3 + 3 = 81 + 3 \). This simplifies to: \( x = 84 \).
04

Verify the Solution

Substitute \( x = 84 \) back into the original equation to verify: \( \sqrt{84-3} = 9 \) This simplifies to: \( \sqrt{81} = 9 \) Since \( 9 = 9 \) is true, the solution is verified.

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Key Concepts

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

square root equations
In this exercise, we're given a square root equation: \( \sqrt{x-3} = 9 \). A square root equation is an equation in which a variable is under a square root. Solving these types of equations often involves removing the square root to isolate the variable.
Remember that not all square root equations have solutions that are real numbers. If the number inside the square root (called the radicand) is negative, then the square root is not defined in the real number system.
algebraic manipulation
Algebraic manipulation is the process of rearranging and simplifying equations to solve for a variable. In this example, we start by squaring both sides of the equation to remove the square root. Here's how it's done:
  • The original equation is \( \sqrt{x-3} = 9 \).
  • Square both sides: \( (\sqrt{x-3})^2 = 9^2 \).
  • This simplifies to: \( x-3 = 81 \).

This step turns our square root equation into a simple linear equation, making it easier to solve for \( x \).
solution verification
After solving the equation, it is crucial to verify the solution by substituting it back into the original equation. This step ensures the solution is correct and satisfies the original problem.
Our solution for the equation was \( x = 84 \). To verify:
  • Substitute \( x = 84 \) back into the original equation: \( \sqrt{84-3} = 9 \).
  • This simplifies to: \( \sqrt{81} = 9 \).
  • Since \( 9 = 9 \) is true, our solution is verified.

Verification is important as it confirms our steps were correct and our algebraic manipulations were accurate.
isolation of variables
Isolating the variable is a fundamental part of solving equations. It involves rearranging the equation such that the variable you are solving for is on one side and everything else is on the other.
In our example:
  • We already had the square root isolated: \( \sqrt{x-3} = 9 \).
  • After squaring both sides and simplifying, we got: \( x - 3 = 81 \).
  • Add 3 to both sides to isolate \( x \): \( x - 3 + 3 = 81 + 3 \).
  • This simplifies to: \( x = 84 \).

Isolating the variable helps us find the solution step-by-step without confusion or errors in the process.

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Most popular questions from this chapter

Without computing the value of each pair of numbers, determine which number is greater. For each problem, explain why. $$3^{-1,600} \text { or } 27^{-500}$$

Challenge In Parts a and b, tell whether the number is rational or irrational. If it is rational, find two integers whose ratio is equal to it. If it is irrational, explain how you know. a. 6\(\sqrt{3}\) b. 2\(\pi\) (Hint: If it is rational, it is equal to \(\frac{a}{b}\) for some integers \(a\) and b. What might \(\pi\) be equal to in terms of \(a\) and \(b ? )\) c. In general, if you multiply a nonzero rational number \(n\) by an irrational number \(m,\) will your result be rational number \(n\) by an irrational? Explain how you know.

Decide whether the expressions in each pair are equivalent. Explain. $$ \sqrt{32}-\sqrt{18} \text { and } \sqrt{14} $$

Simplify each radical expression. If it is already simplified, say so. $$ \sqrt{17}-\sqrt{30} $$

Rewrite each equation in \(y=m x+b\) form, and tell whether the relationship represented by the equation is increasing or decreasing. $$-\frac{1}{3} x=-4-\frac{2}{3} y$$
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