91Ó°ÊÓ

The distributive property is a fundamental algebraic property used to simplify expressions and solve equations. In our example previously mentioned, the step of applying the distributive property looked like this: \(-5(4s+1)\). The property states that you multiply the term outside the parentheses by each term inside, which translates to \(-5 \times 4s = -20s\) and \(-5 \times 1 = -5\), resulting in the expression \(-20s - 5\).

This property is particularly useful when dealing with polynomials or combining like terms. It allows for the consolidation of an equation, which sets the stage for further steps necessary to isolate the variable and solve the inequality.

For those who are just learning this concept, remember: distribution over addition involves multiplying the outside term by every term inside the parentheses individually, then summing the results. It's a lot like handing out flyers to each house in a neighborhood—each house (or term) gets one.

When dividing both sides of an inequality by a negative number, there's a crucial rule to remember: you must flip the inequality sign. This occurs in step 3 of our exercise when dividing by -20, which prompts the transition from \(-20s < 28\) to \(s > -1.4\). This sign change is necessary to maintain the truth of the inequality. If this step is overlooked, the solution will be incorrect.

Why is this necessary? Because dividing by a negative essentially reverses the order of the numbers. For instance, if \(-2 < 1\), when you divide by -1, you get \(2 > -1\), which maintains the original relationship between the values. Always check your work for this when dividing inequalities by negative numbers, as it's a frequent source of errors for many students. Remember, this rule also applies when multiplying by a negative number.