Solve and graph the inequality $-3 y>15$.
Answer (1)
Divide both sides of the inequality 15"> − 3 y > 15 by − 3 , remembering to flip the inequality sign.
Simplify the inequality to get y < − 5 .
Represent the solution on a number line with an open circle at − 5 and shade the region to the left.
The solution to the inequality is y < − 5 , which is graphically represented on the number line. y < − 5
Explanation
Understanding the Inequality We are given the inequality 15"> − 3 y > 15 . Our goal is to solve for y and represent the solution graphically on a number line.
Dividing by a Negative Number To isolate y , we need to divide both sides of the inequality by − 3 . Remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign. So, dividing both sides by − 3 gives us: − 3 − 3 y < − 3 15
Simplifying the Inequality Simplifying the inequality, we get: y < − 5 This means that y can be any number less than − 5 .
Graphing the Solution To graph this inequality on a number line, we first draw a number line. Then, we locate − 5 on the number line. Since the inequality is y < − 5 (and not y ≤ − 5 ), we use an open circle at − 5 to indicate that − 5 is not included in the solution. Finally, we shade the region to the left of − 5 to represent all the values of y that satisfy the inequality.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, imagine you're budgeting your expenses. If you want to spend less than $50 on groceries each week, you're dealing with an inequality: your spending must be less than $50. Similarly, speed limits on roads are another example of inequalities, ensuring that vehicles travel at or below a certain speed for safety. Inequalities help define limits and boundaries in various practical situations.
