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An algebraic expression is a combination of numbers, variables, and operators such as addition or multiplication. For example, in the inequality \(\frac{1}{2}(2x + 2b) > \frac{1}{3}(21 + 3b)\), \(2x + 2b\) and \(21 + 3b\) are algebraic expressions. Variables like \(x\) and \(b\) represent unknown values.
To solve problems involving algebraic expressions, you often need to simplify or manipulate them. This can include distributing factors, combining like terms, or isolating variables.

The distributive property is a key concept in algebra. It allows you to multiply a single term by each term within a parenthesis. For example, \(a(b + c) = ab + ac\). In the given inequality: \[ \frac{1}{2}(2x + 2b) > \frac{1}{3}(21 + 3b) \],
we use the distributive property to simplify. For the left side: \[ \frac{1}{2}(2x + 2b) = \frac{1}{2} \times 2x + \frac{1}{2} \times 2b = x + b \] And for the right side: \[ \frac{1}{3}(21 + 3b) = \frac{1}{3} \times 21 + \frac{1}{3} \times 3b = 7 + b \] This simplification results in a more manageable inequality: \[ x + b > 7 + b \]. Mastering the distributive property aids in numerous mathematical situations, making complex expressions easier to work with.

Isolating the variable is a fundamental step in solving equations and inequalities. It involves rearranging the equation so that the variable is alone on one side of the equation or inequality. In the example \((x + b) > (7 + b)\):
To isolate \(x\), you subtract \(b\) from both sides, resulting in the following steps: \[ (x + b) - b > (7 + b) - b \] This simplifies to: \[ x > 7 \] By removing \(b\) from both sides, we've successfully isolated \(x\). This technique is crucial for finding the value or range of values for a variable.

Inequalities show the relationship between two expressions that are not equal. They use symbols such as \(>\) (greater than), \(<\) (less than), \(\geq\) (greater than or equal to), and \(\leq\) (less than or equal to). For instance, in \[ x + b > 7 + b \], we are expressing that the left side is greater than the right side.
Solving inequalities often includes similar steps to solving equations, like isolating variables and simplifying expressions. After isolating \(x\) in the original problem, we find \[ x > 7 \]. This tells us that \(x\) must be greater than 7. Always remember to consider if multiplying or dividing by a negative number, which reverses the inequality sign.