Solving Inequalities: Find X In 5x + 2 > 17

Nov 13, 2025 by Alex Johnson 44 views

Let's dive into solving a simple inequality. Inequalities are like equations, but instead of an equals sign, they use symbols like '>', '<', '≥', or '≤' to show that one side is greater than, less than, greater than or equal to, or less than or equal to the other side. Today, we'll tackle the inequality $5x + 2 > 17$ and find out what values of $x$ make this statement true. Solving inequalities is a fundamental skill in algebra and is used in various real-world applications, from optimizing resources to understanding constraints in mathematical models. Understanding how to manipulate and solve these inequalities provides a solid foundation for more advanced mathematical concepts. The process involves isolating the variable on one side, much like solving equations, but with a few key differences to watch out for, especially when multiplying or dividing by negative numbers. So, let's get started and break down each step to ensure a clear understanding of how to solve this inequality effectively. This skill will not only help in academic settings but also in practical problem-solving scenarios where understanding boundaries and limitations is crucial.

Understanding Inequalities

Before we jump into the solution, let's quickly recap what inequalities are all about. Think of an inequality as a statement that compares two expressions using symbols other than the equals sign. The main inequality symbols are:

> : Greater than
< : Less than
≥ : Greater than or equal to
≤ : Less than or equal to

Inequalities help us define a range of possible values that satisfy a certain condition. For example, if we say $x > 5$ , it means that $x$ can be any number greater than 5, but not including 5 itself. On the other hand, $x ≥ 5$ means $x$ can be any number greater than or equal to 5, including 5. Understanding these nuances is crucial for interpreting and solving inequalities correctly. Inequalities are used extensively in various fields, such as economics, where they help model constraints like budget limitations, and in physics, where they describe ranges of possible outcomes in experiments. Mastering inequalities allows for a more nuanced and accurate understanding of mathematical relationships, providing valuable insights into real-world scenarios. Furthermore, inequalities play a vital role in optimization problems, where the goal is to find the best possible solution within a given set of constraints, making it an indispensable tool in decision-making processes.

Step-by-Step Solution for $5x + 2 > 17$

Now, let's solve the inequality $5x + 2 > 17$ step-by-step. Our goal is to isolate $x$ on one side of the inequality.

Step 1: Subtract 2 from both sides

To start isolating $x$ , we need to get rid of the '+ 2' on the left side. We do this by subtracting 2 from both sides of the inequality:

$5x + 2 - 2 > 17 - 2$

This simplifies to:

$5x > 15$

Subtracting the same value from both sides maintains the balance of the inequality, ensuring that the relationship between the two expressions remains valid. This is a fundamental principle in solving inequalities, similar to how we manipulate equations. The key is to perform the same operation on both sides to preserve the inequality's truth. This step effectively isolates the term containing the variable, bringing us closer to finding the solution. By removing the constant term from the left side, we pave the way for the next step, which involves isolating the variable itself. This methodical approach ensures accuracy and clarity in solving the inequality.

Step 2: Divide both sides by 5

Next, we need to isolate $x$ completely. Since $x$ is being multiplied by 5, we divide both sides of the inequality by 5:

$\\\frac{5x}{5} > \\\frac{15}{5}$

This simplifies to:

$x > 3$

Dividing both sides by a positive number, in this case 5, does not change the direction of the inequality. This is a crucial rule to remember when solving inequalities. If we were to divide by a negative number, we would need to flip the inequality sign. However, since we're dividing by a positive number, we can proceed directly. This step isolates $x$ , revealing the solution to the inequality. The solution $x > 3$ means that any value of $x$ greater than 3 will satisfy the original inequality. This result provides a clear and concise answer, defining the range of possible values for $x$ . This step highlights the importance of understanding the rules of inequality manipulation to arrive at the correct solution.

Solution

So, the solution to the inequality $5x + 2 > 17$ is $x > 3$ . This means that $x$ can be any number greater than 3. For example, 3.0001, 4, 5, 100, or even 1000000 would all satisfy this inequality.

Graphing the Solution

To visualize the solution $x > 3$ , we can represent it on a number line. Draw a number line and mark the number 3. Since $x$ is strictly greater than 3 (not equal to), we use an open circle at 3 to indicate that 3 is not included in the solution. Then, shade the region to the right of 3, indicating that all numbers greater than 3 are part of the solution. This visual representation provides a clear understanding of the range of values that satisfy the inequality.

Importance of Showing Your Work

Showing your work when solving inequalities (and equations) is extremely important. It allows you (or anyone else) to easily follow your thought process and identify any potential errors. This is especially helpful when dealing with more complex inequalities or systems of inequalities. Clear and organized steps make it easier to debug your solution and ensure accuracy. Additionally, showing your work demonstrates a thorough understanding of the underlying concepts and principles. It also facilitates communication and collaboration, allowing others to understand and build upon your work more effectively. In educational settings, instructors often emphasize the importance of showing work to assess not only the correctness of the answer but also the student's understanding of the solution process.

Common Mistakes to Avoid

When solving inequalities, there are a few common mistakes to watch out for:

Forgetting to flip the inequality sign: If you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you have $-2x > 6$ , dividing by -2 gives $x < -3$ (note the flipped sign).
Incorrectly applying operations: Make sure you perform the same operation on both sides of the inequality to maintain balance.
Misinterpreting the inequality sign: Be clear on the difference between '>', '<', '≥', and '≤'. An open circle on a number line represents strict inequality (>, <), while a closed circle represents inclusive inequality (≥, ≤).

Avoiding these common mistakes will help you solve inequalities accurately and confidently.

Real-World Applications

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

Budgeting: If you have a budget of $100, and you want to buy some items that cost $x each, you can use the inequality $nx ≤ 100$ to determine how many items you can buy.
Speed Limits: A speed limit on a road is an example of an inequality. If the speed limit is 65 mph, then your speed $s$ must satisfy $s ≤ 65$ .
Temperature Ranges: A recipe might specify that a cake should be baked at a temperature between 350°F and 375°F. This can be expressed as $350 ≤ T ≤ 375$ , where $T$ is the temperature.
Resource Allocation: In manufacturing, inequalities can be used to optimize the allocation of resources, such as materials and labor, subject to certain constraints.
Optimization Problems: In economics and engineering, inequalities are used to define constraints in optimization problems, where the goal is to maximize or minimize a certain objective function subject to these constraints.

These examples illustrate how inequalities are used to model and solve real-world problems across various disciplines.

In conclusion, solving inequalities is a crucial skill with wide-ranging applications. By understanding the basic principles and practicing regularly, you can master this skill and apply it to solve real-world problems effectively. From managing budgets to optimizing resources, inequalities provide a powerful tool for making informed decisions and achieving desired outcomes. Keep practicing and exploring different types of inequalities to enhance your problem-solving abilities.

For further learning and practice, visit Khan Academy's inequalities section.