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Chapter 5: Problem 2

Explain the steps you should use to solve \(\sqrt{x}+10<15\).

Short Answer

Expert verified
The solution to the inequality \(\sqrt{x}+10<15\) is \(x < 25\).

Step by step solution

01

Isolate the square root

First, subtract 10 from both sides to isolate the square root. This gives us \(\sqrt{x} < 15 - 10\).
02

Simplify the inequality

Simplify the right side of the equation, resulting in \(\sqrt{x} < 5\).
03

Remove the square root

Square both sides of the inequality to eliminate the square root. This gives us \(x < 25\).

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Key Concepts

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

Understanding Square Roots
Square roots are all about finding a number that, when multiplied by itself, gives you the original number.
For instance, the square root of 25 is 5, because when you multiply 5 by itself, you get 25.
This concept is central when you're solving inequalities involving square roots like \(\sqrt{x} + 10 < 15\).
First, you need to isolate the square root, which means getting it by itself on one side of the inequality.
  • The square root symbol \(\sqrt{}\) can seem a bit tricky, but remember, its purpose is just to find that perfect square.
  • Make sure the square root stands alone as the first step in tackling problems like these.
  • This makes it easier to apply further operations to both sides of the inequality.
Breaking Down Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to find the solution.
In the inequality \(\sqrt{x} + 10 < 15\), we start by subtracting 10 from both sides.
This step isolates the square root, giving us \(\sqrt{x} < 5\).
The goal is clear: keep working with the equation until we make \(x\) the subject of our inequality.
  • Always remember to perform the same operation on both sides of the inequality to maintain balance.
  • Think of algebraic manipulation as a series of logical moves to simplify and solve problems efficiently.
  • This approach allows us to turn complex-looking expressions into simpler forms.
Simplifying Inequalities
Once we have \(\sqrt{x} < 5\), the next goal is to simplify further.
To eliminate the square root, we square both sides of the inequality.
This leaves us with \(x < 25\). It鈥檚 important to remember:
  • When squaring both sides, the inequality direction remains the same because we are dealing with positive values.
  • If you were dealing with negative numbers, squaring could affect the inequality's direction.
  • After this step, you have a straightforward inequality that shows the range of possible values for \(x\).
By simplifying the inequality, we find that any \(x\) less than 25 will satisfy the original condition.

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Most popular questions from this chapter

In Exercises 35鈥�46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=-3 \sqrt{\frac{4 x-7}{3}} $$

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