Solve And Graph Inequalities: K + 9 <= 7

When we talk about solving inequalities, we're essentially looking for a range of values that make a statement true, rather than just a single value like in an equation. Let's dive into solving the inequality $k+9 \leq 7$. Our goal is to isolate the variable, $k$, on one side of the inequality sign. To do this, we need to undo the addition of 9. The opposite of adding 9 is subtracting 9. So, we'll subtract 9 from both sides of the inequality. It's super important to remember that whatever operation you do to one side of an inequality, you must do to the other side to keep the inequality balanced. So, we have $(k+9) - 9 \leq 7 - 9$. On the left side, the $+9$ and $-9$ cancel each other out, leaving us with just $k$. On the right side, $7 - 9$ equals $-2$. Therefore, our inequality simplifies to $k \leq -2$. This means that any number less than or equal to $-2$ will satisfy the original inequality. We've successfully solved the inequality, and now we can move on to representing this solution in different ways.

Understanding the Solution Set in Set-Builder Notation

Now that we've found that $k \leq -2$, let's express this solution using set-builder notation. This is a concise way to describe a set by specifying the properties its members must have. For our inequality, the solution set includes all values of $k$ such that $k$ is less than or equal to $-2$. In set-builder notation, we write this as { $k$ | $k \leq -2$ }. Let's break this down: the curly braces {} indicate that we are defining a set. The variable $k$ before the vertical bar | represents the elements of the set. The vertical bar itself is read as "such that". Everything after the vertical bar describes the condition that the elements must meet. So, { $k$ | $k \leq -2$ } translates to "the set of all $k$ such that $k$ is less than or equal to $-2$". This notation is incredibly useful when dealing with infinite sets or when the rule for membership is complex. It clearly and unambiguously defines the entire collection of numbers that satisfy our inequality, ensuring we don't miss any possible values. It's a fundamental concept in mathematics for defining collections of objects based on shared characteristics.

Representing the Solution in Interval Notation

Moving on, let's express our solution $k \leq -2$ using interval notation. Interval notation is another way to represent a range of numbers on the number line. For our solution, $k$ can be any number from negative infinity up to and including $-2$. When we use interval notation, we use parentheses () to indicate that an endpoint is not included in the interval, and square brackets [] to indicate that an endpoint is included. Since our inequality is $k \leq -2$, the number $-2$ is included in the solution. Therefore, we use a square bracket [ next to $-2$. For the other end of the interval, we're talking about all numbers going towards negative infinity. Infinity, whether positive or negative, is not a specific number that can be included, so we always use a parenthesis ( with it. So, the interval notation for $k \leq -2$ is $(- \infty, -2]$. This signifies all real numbers starting from negative infinity (which is never included) up to $-2$, with $-2$ itself being part of the set. It’s a very visual and compact way to represent continuous sets of numbers and is widely used in calculus and other higher-level mathematics to define domains, ranges, and solution sets for functions and equations.

Graphing the Solution Set

Finally, let's graph the solution set of $k \leq -2$ on a number line. To do this, we first draw a number line and mark a few key points, including $0$ and $-2$. Since our inequality includes $-2$ (because of the "or equal to" part, $\leq$), we will use a closed circle or a filled-in dot at $-2$ on the number line. This closed circle signifies that $-2$ is part of the solution set. If the inequality had been strictly less than ($<$), we would have used an open circle. Now, because $k$ must be less than or equal to $-2$, we need to shade the portion of the number line that represents all numbers less than $-2$. This means we shade to the left of $-2$, extending all the way towards negative infinity. This shaded region, along with the closed circle at $-2$, visually represents every possible value of $k$ that satisfies the inequality $k+9 \leq 7$. Graphing inequalities is a powerful way to visualize the solution set, making it easier to understand the range of values involved. It helps in grasping the concept of continuous sets and how they are represented on a one-dimensional line. Remember, the shading indicates all the numbers that work, and the dot at the endpoint tells you whether that specific endpoint is included or excluded.

Summary of Solutions

To recap our work on the inequality $k+9 \leq 7$:

• Solved Inequality: $k \leq -2$
• Set-Builder Notation: { $k$ | $k \leq -2$ }
• Interval Notation: $(- \infty, -2]$
• Graph: A number line with a closed circle at $-2$ and shading to the left.

These different representations all describe the same set of solutions. Understanding how to move between them – from solving algebraically to expressing in notation and then visualizing on a graph – is a core skill in mathematics. It allows for clear communication of mathematical ideas and provides different perspectives on the same problem. Each method complements the others, offering a robust understanding of inequalities.

For further exploration into inequalities and their solutions, you can visit Khan Academy, a fantastic resource for mathematics learning. Another excellent source of information is Math is Fun, which offers clear explanations and interactive tools for understanding various math concepts.