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Chapter 8: Problem 10

Solve each radical equation. $$\sqrt{5 x-6}=8$$

Short Answer

Expert verified
The solution to the equation is \(x = 14\).

Step by step solution

01

Isolate the square root

To get started, the square root should be alone on one side of the equation, which it already is: \(\sqrt{5x - 6} = 8\)
02

Square both sides

To remove the square root, square both sides of the equation: \((\sqrt{5x - 6})^2 = 8^2\). This gives us \(5x - 6 = 64\).
03

Solve for x

Next, isolate x. Add 6 to both sides of the equation: \(5x = 70\). Then, divide each side by 5: \(x = 14\).

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Key Concepts

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

Solving Equations
Solving equations is a fundamental skill in mathematics. It's all about finding the value of the variable that makes the equation true. Each equation tells a story; for example, finding how many apples you need to buy to spend exactly your $10 budget.
When working with equations, here are some tips to guide you:
  • Always aim to move terms around to get the variable on one side and the numbers on the other.
  • Perform the same mathematical operation on both sides of the equation to maintain balance.
  • Check your solutions by plugging them back into the original equation to ensure they work.
In the exercise, we solved the equation involving a square root, which added complexity. But once the square root was removed (by squaring both sides), it simplified into a straightforward linear equation.
Square Root
The square root is a key concept in mathematics. It refers to a number that, when multiplied by itself, results in the original number. For instance, the square root of 9 is 3 because 3 times 3 equals 9.
Square roots are often many students encounter in various forms of problems, especially radical equations like in our exercise. Solving an equation with a square root often requires isolating the square root on one side, as was done here. Once isolated, you can square both sides to "neutralize" the square root.
This makes it easier to deal with, since it transforms the equation into one without radicals, making it more straightforward to then solve for the variable.
Isolating Variable
Isolating the variable means getting the variable all by itself on one side of the equation. This concept is pivotal because it lets you find out what the variable equals.
  • Begin by looking at the equation to identify terms involving the variable.
  • Use addition, subtraction, multiplication, or division to eliminate other terms from around the variable.
  • Each move should focus on simplifying the equation step by step.
In our exercise, after removing the square root, we isolated the variable 'x' by performing operations to both sides of the equation: adding 6 and then dividing by 5. Remember, isolating the variable is like peeling back layers to reveal the core, which is the variable's value.

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Most popular questions from this chapter

Make Sense? In Exercises \(90-93,\) determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I rationalized a numerical denominator and the simplified denominator still contained an irrational number.

$$\text { Simplify: } \sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}}$$

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$(-64)^{-\frac{2}{3}}$$

Describe what it means to rationalize a denominator. Use both \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) in your explanation.

Use substitution to determine if \(1+\sqrt{3}\) is a solution of \((x-1)^{2}=5\).
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