Solve For K: Inequality 4(k+8) > -4

When we're asked to solve for $k$ in an inequality like $4(k+8)>-4$, our goal is to isolate the variable $k$ on one side of the inequality sign. This means we want to get $k$ all by itself, just like we would if we were solving a regular equation. The process is very similar to solving equations, but we need to be extra careful when multiplying or dividing both sides by a negative number, as this flips the direction of the inequality sign. In this particular problem, we're dealing with a straightforward linear inequality, and we'll walk through each step to ensure we understand how to manipulate it correctly. We'll begin by simplifying the expression on the left side of the inequality, then we'll work on getting the term with $k$ by itself, and finally, we'll isolate $k$ completely. This problem serves as a great introduction to solving inequalities, and by the end, you'll feel much more confident in tackling similar problems.

Our inequality is $4(k+8)>-4$. The first step to solve for $k$ is to simplify the left side of the inequality. We can do this by distributing the $4$ to both terms inside the parentheses. So, $4$ multiplied by $k$ is $4k$, and $4$ multiplied by $8$ is $32$. This gives us the expression $4k + 32$. Now, our inequality looks like this: $4k + 32 > -4$. This step is crucial because it removes the parentheses and makes the inequality easier to work with. Think of it as tidying up the expression before we start moving terms around. It's important to remember the distributive property of multiplication over addition, which states that $a(b+c) = ab + ac$. Applying this property here, with $a=4$, $b=k$, and $c=8$, we correctly get $4k + 32$. This simplified form is what we will use for all subsequent steps in solving for $k$. Always ensure that you apply the distribution accurately, multiplying the outside number by every term inside the parentheses.

Now that we have the simplified inequality $4k + 32 > -4$, our next step to solve for $k$ is to isolate the term containing $k$, which is $4k$. To do this, we need to get rid of the $+32$ on the left side. We can achieve this by subtracting $32$ from both sides of the inequality. Remember, whatever operation we perform on one side of the inequality, we must perform the same operation on the other side to maintain the balance. So, we subtract $32$ from the left side: $(4k + 32) - 32 = 4k$. And we subtract $32$ from the right side: $-4 - 32 = -36$. Since we are subtracting a number, the direction of the inequality sign does not change. Our inequality now becomes $4k > -36$. This step effectively moves the constant term to the right side, bringing us closer to isolating $k$. It's like clearing the path so that $k$ can stand alone. This subtraction is a fundamental inverse operation to the addition we performed during the distribution step, thus undoing it and helping us progress toward our goal of solving for $k$.

The final step to solve for $k$ is to isolate $k$ completely. We currently have $4k > -36$. Since $k$ is being multiplied by $4$, we need to perform the inverse operation, which is division. We will divide both sides of the inequality by $4$. So, on the left side, $4k / 4 = k$. On the right side, we have $-36 / 4$. When we divide $-36$ by $4$, we get $-9$. Since we are dividing by a positive number ($4$), the direction of the inequality sign remains the same. Therefore, our final solution is $k > -9$. This means that any value of $k$ that is greater than $-9$ will satisfy the original inequality. To verify this, we could pick a number greater than $-9$, say $0$, and substitute it back into the original inequality: $4(0+8) > -4$, which simplifies to $4(8) > -4$, or $32 > -4$. This is true. Now let's pick a number less than or equal to $-9$, say $-10$: $4(-10+8) > -4$, which simplifies to $4(-2) > -4$, or $-8 > -4$. This is false, confirming our solution.

Understanding how to solve for $k$ in inequalities is a fundamental skill in algebra. We've seen that by using inverse operations and carefully considering the effect of multiplication and division on the inequality sign, we can systematically find the range of values for $k$ that satisfy the given condition. The inequality $4(k+8) > -4$ led us to the solution $k > -9$. This means that any number larger than $-9$ will make the original statement true. For instance, if $k=0$, $4(0+8) = 32$, and $32$ is indeed greater than $-4$. If $k=-8$, $4(-8+8) = 4(0) = 0$, and $0$ is greater than $-4$. If we try a value less than $-9$, like $k=-10$, we get $4(-10+8) = 4(-2) = -8$, and $-8$ is not greater than $-4$. This confirms our solution set. Remember that the key principles are to simplify, isolate the variable term, and then isolate the variable itself, always keeping the inequality sign's behavior in mind. Practice with various types of inequalities, including those with negative coefficients or requiring division by negative numbers, will further solidify your understanding. The ability to manipulate and solve these equations is a stepping stone to more complex mathematical concepts and problem-solving scenarios.

For further exploration on solving inequalities, you can visit Khan Academy's Algebra Section.