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The distributive property is an essential algebraic concept that allows you to simplify expressions. When you see an expression such as \(7(x+1)\), the distributive property lets you distribute, or spread, the \(7\) to each term inside the parentheses. This is done by multiplying \(7\) by both \(x\) and \(1\), resulting in \(7x + 7\). This step is crucial because it helps to eliminate parentheses and simplify expressions.

In this example, applying the distributive property transforms the inequality from \(2x + 10 \geq 7(x+1)\) to \(2x + 10 \geq 7x + 7\). Remember, always perform the multiplication carefully and watch out for any signs that need to be considered during distribution.

• The number outside the parentheses multiplies each term inside the parentheses.
• This serves to simplify the inequality by removing parentheses.
• Be mindful of positive and negative signs when applying the property.

Combining like terms is an essential skill in simplifying algebraic expressions and inequalities. "Like terms" refer to terms that have the same variable raised to the same power. For example, \(2x\) and \(7x\) are like terms because they both contain the same variable, \(x\), and are to the first power.

In the example inequality \(2x + 10 \geq 7x + 7\), you need to consolidate the "x" terms on one side. By subtracting \(7x\) from both sides, you efficiently combine like terms, yielding \(-5x + 10 \geq 7\).

• Identify terms that have the same variable and power.
• Add or subtract these terms across the inequality to simplify.
• Be meticulous with negative or positive signs to ensure accuracy.

The goal of isolating the variable is to solve for the variable, typically represented by \(x\), so that you can determine its possible values. To isolate the variable in our inequality, \(-5x + 10 \geq 7\), you start by removing any constants from the side with the variable. This involves subtracting \(10\) from both sides, leading to \(-5x \geq -3\).

Next, you need to handle the coefficient next to \(x\). Since it is \(-5\), divide both sides by \(-5\). Remember one critical rule: Whenever you multiply or divide by a negative number, you must flip the inequality sign. Therefore, \(-5x \geq -3\) over \(-5\) results in \(x \leq \frac{3}{5}\).

• Remove constants from the side with the variable by using addition or subtraction.
• Divide or multiply to resolve the coefficient of the variable, paying close attention to the inequality sign.
• Don't forget to flip the inequality sign when dividing or multiplying by a negative number.