Interval Notation Explained: (x−3)(x+7)≥0

Let's understand how to solve inequalities using interval notation.We'll start with this inequality: the product of x minus 3 and x plus 7 is greater than or equal to zero.This means we need to find all values of x where this product is either positive or zero.To solve this, we first need to find the critical points - the values of x where each factor equals zero.For the first factor, x minus 3 equals zero gives us x equals 3.For the second factor, x plus 7 equals zero gives us x equals negative 7.Let's plot these points on a number line to visualize our solution.Here's negative 7, our first critical point.And here's positive 3, our second critical point.These critical points are crucial because they divide our number line into three distinct regions that we'll need to analyze.Now that we have our critical points identified, we're ready to analyze each region of the number line.Now that we have our critical points at negative seven and three, let's analyze the signs in each region.We'll divide our number line into three regions and test a point in each region.In Region 1, where x is less than negative seven, let's test x equals negative eight.In Region 2, between negative seven and three, let's test x equals zero.In Region 3, where x is greater than three, let's test x equals four.Let's summarize our findings with a sign chart.When both factors are negative, their product is positive. When one factor is negative and one is positive, their product is negative. When both factors are positive, their product is positive.Now that we understand the sign pattern in each region, we can write our solution in interval notation.