Solving $4x - 7 \leq 13$ Express In Interval Notation

Jul 5, 2025 by Admin 54 views

$Solving the Inequality $4x - 7 \leq 13$ and Expressing the Solution in Interval Notation$

Introduction

In this article, we will walk through the process of solving the linear inequality $4x - 7 \leq 13$ . Solving inequalities is a fundamental concept in algebra, and it's crucial for various applications in mathematics and real-world scenarios. We will demonstrate each step clearly and express the final solution in interval notation, a standard way of representing sets of real numbers. This notation is particularly useful for inequalities as it concisely shows the range of values that satisfy the given condition. Furthermore, we will explore the meaning of the solution, how it translates graphically on a number line, and why interval notation is the preferred method for expressing such solutions. Understanding these concepts is essential for anyone studying algebra, precalculus, or related fields. By the end of this article, you will have a solid grasp of how to solve linear inequalities and represent their solutions effectively using interval notation. We will also address common mistakes and provide additional examples to ensure a comprehensive understanding. The ability to solve inequalities is not just a mathematical skill but also a valuable tool for problem-solving in various disciplines, including economics, engineering, and computer science. Therefore, mastering this topic is a worthwhile endeavor for anyone seeking a strong foundation in quantitative reasoning.

Step-by-Step Solution

To solve the inequality $4x - 7 \leq 13$ , we will follow a series of algebraic steps similar to solving equations, with one key difference to keep in mind: when multiplying or dividing by a negative number, we must reverse the inequality sign. Here's a detailed breakdown of the solution:

Isolate the term with x:

Our first goal is to isolate the term containing the variable x. In this case, it's 4x. To do this, we need to eliminate the constant term (-7) on the left side of the inequality. We can achieve this by adding 7 to both sides of the inequality. This maintains the balance of the inequality and moves us closer to isolating x.
```
4x - 7 + 7 \leq 13 + 7
```
This simplifies to:
```
4x \leq 20
```
Solve for x:

Now that we have isolated the 4x term, the next step is to isolate x itself. Since x is being multiplied by 4, we need to undo this multiplication by performing the inverse operation, which is division. We divide both sides of the inequality by 4. Since 4 is a positive number, we do not need to reverse the inequality sign.
```
\frac{4x}{4} \leq \frac{20}{4}
```
This simplifies to:
```
x \leq 5
```
Express the solution in interval notation:

The solution $x \leq 5$ means that x can be any real number that is less than or equal to 5. To express this in interval notation, we use a combination of parentheses and brackets. A bracket indicates that the endpoint is included in the interval, while a parenthesis indicates that the endpoint is not included. Since our inequality includes "equal to," we will use a bracket for the endpoint 5. The solution extends indefinitely to the left (negative infinity), which we represent with $-\infty$ . Infinity is not a number, so we always use a parenthesis with infinity.

Therefore, the interval notation for $x \leq 5$ is:
```
(-\infty, 5]
```
This notation reads as "all real numbers from negative infinity up to and including 5."

Understanding Interval Notation

Interval notation is a standardized way to represent sets of real numbers. It uses parentheses and brackets to indicate whether the endpoints are included in the set. Understanding interval notation is crucial for accurately expressing solutions to inequalities and for working with sets of numbers in various mathematical contexts. The key components of interval notation are:

: Parentheses ( ): Parentheses indicate that the endpoint is not included in the interval. For example, $(a, b)$ represents all real numbers between a and b, excluding a and b.
: Brackets [ ]: Brackets indicate that the endpoint is included in the interval. For example, $[a, b]$ represents all real numbers between a and b, including a and b.
: Infinity Symbols $\infty$ and $-\infty$ : Infinity symbols are used to represent intervals that extend indefinitely in one or both directions. Since infinity is not a specific number, it is always enclosed in parentheses. For example, $(-\infty, a]$ represents all real numbers less than or equal to a, and $[b, \infty)$ represents all real numbers greater than or equal to b.

Let's illustrate with some examples:

: The interval (-3, 2) represents all real numbers strictly between -3 and 2. This includes numbers like -2.99, 0, 1, and 1.99, but it does not include -3 or 2.
: The interval [1, 5] represents all real numbers between 1 and 5, including 1 and 5. This includes numbers like 1, 2, 3, 4, and 5.
: The interval [0, $\infty$ ) represents all non-negative real numbers, including 0 and extending to positive infinity. This includes numbers like 0, 1, 2, 3, and so on.
: The interval $(-\infty$ , -2) represents all real numbers less than -2. This includes numbers like -3, -4, -5, and so on.

Union of Intervals:

Sometimes, the solution to an inequality may consist of two or more separate intervals. In such cases, we use the union symbol "U" to combine these intervals. For example, if the solution is $x < 1$ or $x > 3$ , we would express it in interval notation as $(-\infty, 1) U (3, \infty)$ . This represents all real numbers less than 1, as well as all real numbers greater than 3.

Understanding how to use and interpret interval notation is essential for conveying mathematical solutions accurately and concisely. It provides a clear way to represent sets of numbers and is a fundamental tool in various branches of mathematics.

Graphical Representation

The solution to an inequality can also be represented graphically on a number line. This visual representation provides an intuitive understanding of the solution set. To graph the solution $x \leq 5$ , we follow these steps:

Draw a number line: Draw a horizontal line and mark several points on it, including 0 and 5. The number line extends infinitely in both directions, representing all real numbers.
Mark the endpoint: Since our inequality includes "equal to" ( $x \leq 5$ ), we use a closed circle (or a filled-in dot) at 5 to indicate that 5 is included in the solution. If the inequality were strict (e.g., $x < 5$ ), we would use an open circle to indicate that 5 is not included.
Shade the appropriate region: The inequality $x \leq 5$ means that x can be any number less than or equal to 5. Therefore, we shade the region of the number line to the left of 5, including 5 itself. This shaded region represents all the values of x that satisfy the inequality.

Interpreting the Graph:

The graph clearly shows all the real numbers that satisfy the inequality. The closed circle at 5 indicates that 5 is part of the solution, and the shaded region extending to the left indicates that all numbers less than 5 are also solutions. If we had the inequality $x > 2$ , we would use an open circle at 2 and shade the region to the right, indicating that all numbers greater than 2 are solutions, but 2 itself is not.

Using the Graph to Write Interval Notation:

The graphical representation also helps in writing the solution in interval notation. The shaded region extending from negative infinity to 5, including 5, corresponds to the interval $(-\infty, 5]$ . The left parenthesis indicates that negative infinity is not included, and the right bracket indicates that 5 is included.

Union of Intervals on a Number Line:

If the solution consists of a union of intervals, the graph will have multiple shaded regions. For example, if the solution were $x < 1$ or $x > 3$ , we would have two shaded regions: one to the left of 1 (with an open circle at 1) and one to the right of 3 (with an open circle at 3). The corresponding interval notation would be $(-\infty, 1) U (3, \infty)$ .

In summary, graphing inequalities on a number line provides a visual aid to understanding the solution set and helps in accurately expressing the solution in interval notation. The graph makes it clear which numbers are included in the solution and which are not, and it is a valuable tool for solving and interpreting inequalities.

Common Mistakes

When solving inequalities and expressing the solutions in interval notation, several common mistakes can occur. Being aware of these potential pitfalls can help you avoid them and ensure accuracy. Here are some of the most frequent errors:

Forgetting to reverse the inequality sign:

The most critical rule to remember when solving inequalities is that you must reverse the inequality sign whenever you multiply or divide both sides by a negative number. For example, if you have the inequality $-2x < 6$ , dividing both sides by -2 requires you to change the "less than" sign to a "greater than" sign, resulting in $x > -3$ . Forgetting this rule is a common mistake that leads to incorrect solutions.
Incorrectly using parentheses and brackets:

Using parentheses and brackets correctly in interval notation is essential for accurately representing the solution set. Remember that parentheses ( ) indicate that the endpoint is not included, while brackets [ ] indicate that the endpoint is included. For example, the interval $(2, 5]$ includes all numbers greater than 2 and less than or equal to 5. Confusing these symbols can lead to misinterpretations of the solution.
Misinterpreting infinity symbols:

Infinity symbols ( $\infty$ and $-\infty$ ) represent unbounded intervals. Always use a parenthesis with infinity symbols because infinity is not a specific number and cannot be included in an interval. Writing something like $[\infty, 5]$ is incorrect because it implies that infinity is a specific endpoint.
Errors in algebraic manipulation:

Simple algebraic errors, such as incorrect addition, subtraction, multiplication, or division, can lead to wrong solutions. It's essential to double-check each step in your algebraic manipulation to ensure accuracy. For example, if you're solving $3x + 5 \leq 14$ , make sure you correctly subtract 5 from both sides before dividing by 3.
Not understanding the union of intervals:

When the solution to an inequality involves a union of intervals (using the "U" symbol), it means that the solution consists of multiple separate intervals. Make sure you correctly identify and represent each interval. For example, if the solution is $x < -1$ or $x > 2$ , the correct interval notation is $(-\infty, -1) U (2, \infty)$ .
Difficulty with compound inequalities:

Compound inequalities, such as $a < x < b$ or $x < a$ or $x > b$ , require careful attention. Make sure you understand the meaning of the inequality symbols and how they combine to define the solution set. For instance, $2 < x \leq 5$ means that x is greater than 2 but less than or equal to 5.

By being mindful of these common mistakes, you can improve your accuracy in solving inequalities and expressing their solutions in interval notation. Practice and careful attention to detail are key to mastering this topic.

Conclusion

In this article, we have thoroughly explored the process of solving the inequality $4x - 7 \leq 13$ and expressing the solution in interval notation. We began by detailing the step-by-step algebraic manipulation required to isolate x, ensuring that we followed the crucial rule of reversing the inequality sign when dividing by a negative number. The solution, $x \leq 5$ , was then translated into the interval notation $(-\infty, 5]$ , which succinctly represents all real numbers less than or equal to 5.

We also delved into the concept of interval notation itself, emphasizing the importance of parentheses and brackets in indicating whether endpoints are included in the solution set. The use of infinity symbols was explained, and examples were provided to clarify the notation for various types of intervals. The ability to understand and use interval notation is a fundamental skill in mathematics, enabling clear and concise communication of solution sets.

Furthermore, we discussed the graphical representation of the solution on a number line. This visual aid offers an intuitive understanding of the solution, showing how the shaded region corresponds to the set of numbers that satisfy the inequality. The graphical approach reinforces the concept of interval notation and helps in visualizing the range of possible values for x.

Finally, we addressed common mistakes that students often make when solving inequalities, such as forgetting to reverse the inequality sign or misusing parentheses and brackets. By highlighting these pitfalls, we aim to help readers avoid errors and develop a more robust understanding of the topic. The ability to recognize and correct mistakes is a critical aspect of mathematical proficiency.

Mastering the solution of inequalities and the use of interval notation is essential for success in algebra and beyond. The concepts and techniques covered in this article provide a solid foundation for more advanced mathematical topics, such as calculus and real analysis. By practicing these skills and paying attention to detail, students can confidently tackle a wide range of inequality problems.

The solution to the inequality $4x - 7 \leq 13$ is indeed $x \in (-\infty, 5]$ . This demonstrates the power and precision of interval notation in expressing mathematical solutions.