91Ó°ÊÓ

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). In algebra, cube roots are often represented as \(a^{1/3}\) where 'a' is the number.
In the given problem, we have cube roots like \( (2x + 5)^{1/3}\) and \( (6x - 1)^{1/3}\). These represent the cube roots of the expressions inside the parentheses.
To solve equations involving cube roots, one method used is to ‘isolate’ one of the cube root terms. This means you'll try to get one of the cube root terms by itself on one side of the equation.
Once isolated, to clear the cube root, you can cube both sides of the equation. This means raising each side of the equation to the power of three because cubing a cube root will leave you with the original expression inside. For instance, in our exercise, \[ ( (2x + 5)^{1/3})^3 = ( (6x - 1)^{1/3})^3 \] simplifies to \[ 2x + 5 = 6x - 1 \].

A linear equation is an equation between two variables that forms a straight line when plotted on a graph. The general form of a linear equation in one variable is \(ax + b = c\), where 'a', 'b', and 'c' are constants.
In the solved exercise, after cubing both sides of the equation to remove the cube roots, we are left with \(2x + 5 = 6x - 1\). This is a straightforward linear equation.
To solve it, we use basic algebraic operations:

• Subtract 2x from both sides to find: \[ 5 = 4x - 1 \]
• Next, add 1 to both sides: \[ 6 = 4x \]
• Finally, divide both sides by 4 to isolate x: \[ x = \frac{6}{4} = \frac{3}{2} \]

So, solving linear equations generally involves isolating the variable on one side of the equation by using addition, subtraction, multiplication, or division.

Verification of a solution means substituting the solution back into the original equation to check if it satisfies the equation.
This step ensures that no mistakes were made during solving.
In our example, after finding \ ( x = \frac{3}{2} )\, we substitute \(x\) back into the original equation \[ ( 2x+5 )^{1/3} - ( 6x-1 )^{1/3} = 0\]
Substitute \( x = \frac{3}{2}\): \[ ( 2(\frac{3}{2}) + 5 )^{1/3} - ( 6(\frac{3}{2}) - 1 )^{1/3} = 0 \]
Simplify within the cube roots: \[ ( 3 + 5 )^{1/3} - ( 9 - 1 )^{1/3} = 0 \] \[ 8^{1/3} - 8^{1/3} = 0 \] \[ 2 - 2 = 0 \]
The left-hand side equals the right-hand side, confirming that \( x = \frac{3}{2} \) is indeed the correct solution.