Solving The Inequality: 2(4+2x) >= 5x+5

Welcome, math enthusiasts! Today, we're diving deep into the world of inequalities, specifically tackling the problem: Solve the inequality $2(4+2 x) \geq 5 x+5$. Inequalities are fundamental tools in mathematics, allowing us to express relationships where values are not necessarily equal but fall within a certain range. They are used extensively in various fields, from economics and engineering to computer science and everyday decision-making. Understanding how to manipulate and solve inequalities is a crucial skill, and this article will guide you step-by-step through the process, ensuring you grasp the concepts with clarity and confidence. We'll break down each part of the inequality, simplify it, and arrive at the solution set, which represents all the possible values of 'x' that satisfy the given condition. So, grab your thinking caps, and let's get started on this mathematical journey!

Understanding the Inequality: $2(4+2 x) \geq 5 x+5$

Before we embark on the journey to solve the inequality $2(4+2 x) \geq 5 x+5$, let's take a moment to appreciate the components that make up this mathematical statement. At its core, an inequality is a comparison between two expressions. Unlike an equation, which uses an equals sign (=) to state that two expressions have the same value, an inequality uses symbols like 'greater than' (>), 'less than' (<), 'greater than or equal to' (≥), or 'less than or equal to' (≤) to show a relationship of magnitude. In our specific case, the symbol '≥' tells us that the expression on the left side must be greater than or equal to the expression on the right side. The expression on the left is $2(4+2 x)$, which involves multiplication and addition within parentheses. The expression on the right is $5 x+5$, a simpler linear expression. Our goal is to find all the values of the variable 'x' that make this statement true. This involves isolating 'x' on one side of the inequality, much like we would solve an equation, but with a few key considerations regarding how certain operations can affect the direction of the inequality sign. We'll be using fundamental algebraic properties to simplify and rearrange the inequality, ultimately revealing the set of solutions.

Step-by-Step Solution: Simplifying the Inequality

The first crucial step to solve the inequality $2(4+2 x) \geq 5 x+5$ involves simplifying both sides of the inequality. On the left side, we have the expression $2(4+2 x)$. To simplify this, we apply the distributive property, which means we multiply the number outside the parentheses (2) by each term inside the parentheses (4 and 2x). So, $2 \times 4$ equals 8, and $2 \times 2x$ equals $4x$. Therefore, the left side of the inequality simplifies to $8 + 4x$. The right side of the inequality, $5 x+5$, is already in its simplest form, so no further simplification is needed there. After applying the distributive property, our inequality now reads: $8 + 4x \geq 5 x+5$. This simplified form makes it much easier to proceed with isolating the variable 'x'. Remember, the goal is to gather all terms containing 'x' on one side of the inequality and all constant terms on the other. This process involves a series of algebraic manipulations that preserve the truth of the inequality, provided we handle certain operations with care.

Isolating the Variable 'x'

Now that we have simplified the inequality to $8 + 4x \geq 5 x+5$, our next objective is to solve the inequality $2(4+2 x) \geq 5 x+5$ by isolating the variable 'x'. To do this, we'll perform a series of operations on both sides of the inequality. A common strategy is to move all the 'x' terms to one side and all the constant terms to the other. Let's start by moving the 'x' terms. We can subtract $4x$ from both sides of the inequality. This is a valid operation because subtracting the same value from both sides does not change the direction of the inequality sign. So, we have: $8 + 4x - 4x \geq 5 x + 5 - 4x$. This simplifies to $8 \geq x + 5$. Now, we need to isolate 'x' completely. To do this, we subtract 5 from both sides of the inequality: $8 - 5 \geq x + 5 - 5$. This gives us $3 \geq x$.

Alternatively, we could have chosen to move the 'x' terms to the left side. Let's explore that path: starting again from $8 + 4x \geq 5 x+5$, we subtract $5x$ from both sides: $8 + 4x - 5x \geq 5 x + 5 - 5x$. This simplifies to $8 - x \geq 5$. Now, to isolate the '-x' term, we subtract 8 from both sides: $8 - x - 8 \geq 5 - 8$. This results in $-x \geq -3$. At this point, we have isolated '-x', but we need to find the value of 'x', not '-x'. This brings us to a crucial rule in inequality manipulation: when multiplying or dividing both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. So, multiplying both sides by -1 and reversing the sign, we get: $(-1)(-x) \leq (-1)(-3)$, which simplifies to $x \leq 3$. Notice that both methods, when executed correctly, lead to the same solution.

Interpreting the Solution: $x \leq 3$

We have successfully navigated the process to solve the inequality $2(4+2 x) \geq 5 x+5$, and our result is $x \leq 3$. Now, it's vital to understand what this solution actually means. The statement $x \leq 3$ signifies that any real number that is less than or equal to 3 will satisfy the original inequality. This is not a single, specific value for 'x', but rather an infinite set of values. To visualize this, imagine a number line. The solution set includes the number 3 itself, as indicated by the 'or equal to' part of the symbol (≤). It also includes all the numbers to the left of 3 – numbers like 2, 1, 0, -1, -1.5, -100, and so on, extending infinitely in the negative direction. The set of numbers that satisfy this inequality is called the solution set, and it can be expressed in various ways, including interval notation. In interval notation, $x \leq 3$ is written as $(-\infty, 3]$. The parenthesis next to negative infinity indicates that it's not included (as infinity is not a number), and the square bracket next to 3 indicates that 3 itself is included in the solution set. This understanding is crucial for applying inequalities in real-world scenarios, where we often deal with ranges of possibilities rather than exact figures. For example, if a certain condition is met when 'x' is less than or equal to 3, it means any value up to and including 3 is acceptable.

Verification: Checking Our Solution

To ensure that our work to solve the inequality $2(4+2 x) \geq 5 x+5$ is accurate, we should always perform a verification step. This involves picking test values for 'x' that fall within our solution set ($x \leq 3$) and values that fall outside it, and substituting them back into the original inequality to see if the statement holds true. Let's start with a value that is within our solution set, for example, $x=2$. Substituting $x=2$ into the original inequality $2(4+2 x) \geq 5 x+5$:

Left side: $2(4 + 2(2)) = 2(4 + 4) = 2(8) = 16$

Right side: $5(2) + 5 = 10 + 5 = 15$

Since $16 \geq 15$, the inequality holds true for $x=2$. This supports our solution. Now, let's pick a value that is equal to the boundary, $x=3$.

Left side: $2(4 + 2(3)) = 2(4 + 6) = 2(10) = 20$

Right side: $5(3) + 5 = 15 + 5 = 20$

Since $20 \geq 20$, the inequality holds true for $x=3$, confirming that the boundary is included. Finally, let's test a value that is outside our solution set, for example, $x=4$.

Left side: $2(4 + 2(4)) = 2(4 + 8) = 2(12) = 24$

Right side: $5(4) + 5 = 20 + 5 = 25$

Since $24 \geq 25$ is false, the inequality does not hold true for $x=4$. This confirms that values greater than 3 are not part of our solution. This verification process gives us strong confidence in our derived solution, $x \leq 3$.

Conclusion: Mastering Inequalities

In conclusion, we have successfully navigated the process to solve the inequality $2(4+2 x) \geq 5 x+5$, arriving at the solution $x \leq 3$. We've seen that solving inequalities involves many of the same algebraic techniques used for solving equations, with the critical difference being the rule about reversing the inequality sign when multiplying or dividing by a negative number. Understanding this rule is paramount to obtaining the correct solution set. Inequalities are not just abstract mathematical concepts; they are powerful tools that describe relationships and constraints in a vast array of applications. Whether you're optimizing resource allocation in business, defining boundaries in engineering, or analyzing data trends, the ability to effectively solve and interpret inequalities is an invaluable skill. By breaking down the problem, simplifying expressions, isolating the variable, and carefully interpreting the result, you can confidently tackle a wide range of inequality problems. Remember to always verify your solutions by testing values within and outside your proposed solution set. This practice not only confirms your accuracy but also deepens your understanding of the underlying mathematical principles.

For further exploration into the fascinating world of inequalities and algebra, I highly recommend visiting Khan Academy's comprehensive resources on algebra. They offer a wealth of tutorials, practice problems, and in-depth explanations that can further solidify your understanding. Additionally, exploring Wolfram MathWorld can provide advanced insights and definitions related to inequalities and mathematical concepts.