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The distributive property is a fundamental algebraic principle that allows you to simplify expressions by distributing a factor across terms inside parentheses. In the exercise, we see the application of this property with the expression \(4(x + 1) + 2\). The distributive property states that \(a(b + c) = ab + ac\). Here, \(4\) is distributed over \(x + 1\), resulting in \(4 \times x + 4 \times 1\). This simplifies the expression to \(4x + 4\).

Using the distributive property helps us to transform complex expressions into simpler forms. It is essential when dealing with equations and inequalities where terms are grouped in parentheses. Always remember to apply this property correctly to avoid errors in simplification.

Solving inequalities is similar to solving equations, but with one key difference: the inequality signs (such as \(\geq\), \(\leq\), \(>\), and \(<\)). These symbols denote the relationship between two expressions. The main goal of solving an inequality is to find all possible values that make the inequality true.

For the given inequality \(4x + 6 \geq 3x + 6\), we begin by simplifying both sides. Then, we need to isolate the variable \(x\). Unlike equations, when you multiply or divide by a negative number in inequalities, the inequality sign changes direction. However, in this exercise, we do not encounter a negative multiplication or division.

Usually, the solution to an inequality is expressed as a range of values, which in this case is \(x \geq 0\). This means any real number equal to or greater than 0 satisfies the inequality.

Algebraic manipulation involves rearranging and simplifying expressions or equations to achieve a desired form, usually isolating a variable. In the context of the exercise, this means getting \(x\) by itself on one side of the inequality.

To isolate \(x\), we subtract \(3x\) from both sides of \(4x + 6 \geq 3x + 6\), which gives us \(x \geq 0\). Subtraction is a fundamental operation in algebraic manipulation that helps balance the inequality or equation.

• Identify like terms on both sides of the inequality.
• Apply necessary operations such as addition, subtraction, multiplication, or division.
• Simplify the expression as much as possible.

Algebraic manipulation requires careful steps to maintain the balance and integrity of the equation or inequality. With practice, these steps become intuitive, aiding in solving more complex algebraic problems efficiently.