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When solving inequalities, our goal is to find all the values that make the inequality true. Inequalities are similar to equations but instead of an equal sign, they use symbols like \(<\), \(>\), \(\leq\), or \(\geq\). These symbols denote that one side of the inequality is less than, greater than, or equal to the other side.
This distinction means that the solutions to inequalities can describe a range of numbers rather than a specific value. To solve the inequality \(4(2x-1) \geq 0\), we must perform operations to isolate the variable \(x\). Each step must preserve the inequality's direction unless you multiply or divide by a negative number. In such a case, you would reverse the inequality symbol. This careful treatment ensures you find the correct range of solutions for \(x\).
The steps include simplifying expressions, using basic operations, and dividing or multiplying to isolate the variable.

The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations or inequalities. It states that when you multiply a number by a sum or difference, you can distribute the multiplication over each term inside the parentheses. Mathematically, this is written as \(a(b + c) = ab + ac\).
In the given exercise, we apply the distributive property to \(4(2x-1)\). This means we multiply 4 by both \(2x\) and \(-1\), resulting in the expression \(8x - 4\).
Utilizing this property allows us to eliminate parentheses, making it easier to continue solving the inequality. Make sure that every term inside the parentheses gets multiplied by the number outside. This skill is crucial not only for solving inequalities but also for working with polynomials and other algebraic expressions.

Algebraic manipulation involves various techniques to rearrange and simplify expressions or equations to solve for a variable. This process includes operations like adding, subtracting, multiplying, and dividing terms. The objective is to isolate the variable, making it the subject of the equation or inequality.
For the inequality \(8x - 4 \geq 0\), we begin by adding 4 to both sides, resulting in \(8x \geq 4\). This step moves us closer to isolating \(x\).
Next, divide both sides by 8 to solve for \(x\), simplifying the expression to \(x \geq \frac{1}{2}\). Remember, if you multiply or divide by a negative number during manipulation, you must reverse the inequality sign. However, since this operation doesn't involve a negative factor, we retain the direction of \(\geq\). Each operation systematically breaks down the inequality to make \(x\) easier to identify and understand.