Solving Inequalities And Expressing Solutions In Interval Notation

This article delves into the methods of solving inequalities and expressing their solutions using interval notation. We will explore two specific examples, providing a step-by-step guide to understanding and mastering these essential mathematical concepts. This comprehensive guide aims to equip you with the knowledge and skills necessary to tackle a wide range of inequality problems.

1. Solving and Expressing the Solution in Interval Notation: $-3 \leq 2x - 1 < 7$

In this section, we will address the inequality $-3 \leq 2x - 1 < 7$. The primary goal is to isolate the variable x in the middle of the inequality. To achieve this, we will perform a series of algebraic manipulations, ensuring that each operation is applied to all three parts of the inequality. This approach maintains the balance and integrity of the inequality throughout the solving process. First, we begin by adding 1 to all parts of the inequality. This step is crucial for isolating the term containing x. By adding 1, we effectively eliminate the constant term on the left-hand side, bringing us closer to isolating x. This addition transforms the inequality into a simpler form, making it easier to proceed with the subsequent steps. The result of this operation is $-3 + 1 \leq 2x - 1 + 1 < 7 + 1$, which simplifies to $-2 \leq 2x < 8$. This new form of the inequality is more manageable and sets the stage for the next step in isolating x. Now that we've added 1 to all parts, the next logical step is to divide all parts of the inequality by 2. This division is essential for isolating x completely. By dividing by 2, we undo the multiplication that was previously applied to x, bringing us closer to the final solution. This step is a fundamental algebraic operation that preserves the inequality as long as we divide by a positive number, which is the case here. After performing the division, the inequality becomes $-2/2 \leq 2x/2 < 8/2$, which simplifies to $-1 \leq x < 4$. This simplified form provides a clear range for the possible values of x. The inequality $-1 \leq x < 4$ signifies that x is greater than or equal to -1 and strictly less than 4. This means that x can take any value within this range, including -1 but not including 4. This understanding is crucial for expressing the solution in interval notation. Finally, we express the solution in interval notation. Interval notation is a concise way of representing a range of values. It uses brackets and parentheses to indicate whether the endpoints are included or excluded from the solution set. In this case, the solution includes -1, so we use a square bracket on the left side. The solution does not include 4, so we use a parenthesis on the right side. Therefore, the interval notation for the solution is [-1, 4). This notation clearly and succinctly represents all values of x that satisfy the original inequality.

2. Determining the Interval that Satisfies Both $x + 3 > 0$ and $2x extless 10$

In this section, our objective is to determine the interval that satisfies both inequalities: $x + 3 > 0$ and $2x \leq 10$. This involves solving each inequality separately and then finding the intersection of their solution sets. The intersection represents the values of x that satisfy both inequalities simultaneously. This process is fundamental in various mathematical applications, including optimization problems and systems of equations. We will first solve the inequality $x + 3 > 0$. To isolate x, we subtract 3 from both sides of the inequality. This subtraction is a basic algebraic operation that preserves the inequality and helps us to determine the range of values for x. Subtracting 3 from both sides gives us $x + 3 - 3 > 0 - 3$, which simplifies to $x > -3$. This inequality tells us that x must be greater than -3 to satisfy the first condition. The solution $x > -3$ represents an open interval extending from -3 to infinity. This means that any value of x greater than -3 will satisfy the first inequality. It is important to note that -3 itself is not included in the solution set, as the inequality is strictly greater than. This solution forms one part of the overall solution, and we will combine it with the solution of the second inequality to find the final answer. Next, we solve the inequality $2x \leq 10$. To isolate x, we divide both sides of the inequality by 2. This division is another fundamental algebraic operation that helps us to determine the range of values for x. Dividing by 2 gives us $2x/2 \leq 10/2$, which simplifies to $x \leq 5$. This inequality tells us that x must be less than or equal to 5 to satisfy the second condition. The solution $x \leq 5$ represents a closed interval extending from negative infinity up to and including 5. This means that any value of x less than or equal to 5 will satisfy the second inequality. The inclusion of 5 in the solution set is indicated by the "less than or equal to" sign. To find the interval that satisfies both inequalities, we need to find the intersection of the two solution sets. The first inequality gives us $x > -3$, which can be represented in interval notation as $(-3, \infty)$. The second inequality gives us $x \leq 5$, which can be represented in interval notation as $(-\infty, 5]$. The intersection of these two intervals is the set of values that are both greater than -3 and less than or equal to 5. This means we are looking for the values that overlap in both intervals. By visualizing the number line, we can see that the overlapping region starts just after -3 and extends up to and includes 5. Therefore, the interval that satisfies both inequalities is the intersection of $(-3, \infty)$ and $(-\infty, 5]$, which is (-3, 5]. This final interval notation represents the solution set where x is greater than -3 but less than or equal to 5. This is the range of values that satisfy both original inequalities simultaneously.

In conclusion, solving inequalities and expressing their solutions in interval notation is a fundamental skill in mathematics. By understanding the step-by-step processes involved in isolating variables and representing solutions, you can effectively tackle a wide range of problems. The examples provided in this article offer a solid foundation for further exploration and mastery of these concepts.