Solving Inequalities And Representing Solutions On Number Lines

This article delves into the process of solving linear inequalities and representing their solutions on a number line. Linear inequalities are mathematical expressions that use inequality symbols such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤) to compare two values. Solving an inequality involves finding the range of values that satisfy the given condition. This range of values can then be visually represented on a number line, providing a clear understanding of the solution set. The number line representation is crucial for visualizing the infinite number of solutions that an inequality often possesses. This article will walk you through the steps of solving several inequalities and matching them to their corresponding number line representations. This includes simplifying the inequalities, isolating the variable, and correctly interpreting the solution set on the number line. Understanding these concepts is fundamental in algebra and has wide applications in various fields, including economics, physics, and computer science. The following sections will break down each inequality step-by-step, ensuring a clear and comprehensive understanding of the solution process and its graphical representation.

H2: Inequality 1: $-\frac{x}{10}+\frac{1}{5} \geq-\frac{33}{55}$

Let's start by tackling the first inequality: $-\frac{x}{10}+\frac{1}{5} \geq-\frac{33}{55}$. Our primary goal here is to isolate the variable x on one side of the inequality. This involves several algebraic manipulations to simplify the expression and ultimately find the range of values for x that satisfy the inequality. First, we simplify the fraction $-\frac{33}{55}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 11. This gives us $-\frac{3}{5}$. So, the inequality becomes: $-\frac{x}{10}+\frac{1}{5} \geq-\frac{3}{5}$. Next, we want to eliminate the fractions to make the equation easier to work with. We find the least common multiple (LCM) of the denominators 10 and 5, which is 10. Multiplying every term in the inequality by 10 will clear the fractions. This gives us: $10(-\frac{x}{10}) + 10(\frac{1}{5}) \geq 10(-\frac{3}{5})$. Simplifying this, we get: $-x + 2 \geq -6$. Now, we want to isolate the term with x. We subtract 2 from both sides of the inequality: $-x + 2 - 2 \geq -6 - 2$, which simplifies to $-x \geq -8$. It's crucial to remember that when we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign. In this case, we need to multiply both sides by -1 to get x by itself. So, we have: $(-1)(-x) \leq (-1)(-8)$, which simplifies to $x \leq 8$. This solution tells us that x can be any value less than or equal to 8. On a number line, this is represented by a closed circle (or a solid dot) at 8, indicating that 8 is included in the solution, and an arrow extending to the left, indicating that all values less than 8 are also solutions. Understanding this representation is crucial for visualizing the solution set of the inequality.

H2: Inequality 2: $\frac{5 x}{3}-\frac{11}{6} \geq \frac{163}{2}$

Now, let's address the second inequality: $\frac{5 x}{3}-\frac{11}{6} \geq \frac{163}{2}$. Again, our main objective is to isolate x. This involves a series of algebraic steps to simplify the inequality and determine the range of values that satisfy it. The first step is to eliminate the fractions. We identify the least common multiple (LCM) of the denominators 3, 6, and 2. The LCM is 6. Multiplying every term in the inequality by 6 will clear the fractions. This gives us: $6(\frac{5 x}{3}) - 6(\frac{11}{6}) \geq 6(\frac{163}{2})$. Simplifying, we get: $10x - 11 \geq 489$. Next, we want to isolate the term with x. We add 11 to both sides of the inequality: $10x - 11 + 11 \geq 489 + 11$, which simplifies to $10x \geq 500$. To solve for x, we divide both sides of the inequality by 10: $\frac{10x}{10} \geq \frac{500}{10}$, which simplifies to $x \geq 50$. This solution tells us that x can be any value greater than or equal to 50. On a number line, this is represented by a closed circle (or a solid dot) at 50, indicating that 50 is included in the solution, and an arrow extending to the right, indicating that all values greater than 50 are also solutions. Visualizing this on the number line helps to understand the infinite range of solutions for this inequality. Properly solving and representing inequalities like this is a fundamental skill in algebra and provides a solid foundation for more complex mathematical problems.

H2: Inequality 3: $\frac{3 x}{2}+105 \leq 96$

Moving on to the third inequality: $\frac{3 x}{2}+105 \leq 96$, our consistent goal remains to isolate x. This process involves simplifying the inequality through algebraic manipulations to find the values of x that satisfy the condition. First, we want to isolate the term with x. We subtract 105 from both sides of the inequality: $\frac{3 x}{2}+105 - 105 \leq 96 - 105$, which simplifies to $\frac{3 x}{2} \leq -9$. Now, to get rid of the fraction, we multiply both sides of the inequality by 2: $2(\frac{3 x}{2}) \leq 2(-9)$, which simplifies to $3x \leq -18$. To solve for x, we divide both sides of the inequality by 3: $\frac{3x}{3} \leq \frac{-18}{3}$, which simplifies to $x \leq -6$. This solution tells us that x can be any value less than or equal to -6. On a number line, this is represented by a closed circle (or a solid dot) at -6, indicating that -6 is included in the solution, and an arrow extending to the left, indicating that all values less than -6 are also solutions. The number line representation provides a visual understanding of the infinite solutions that satisfy the inequality. Understanding how to solve and represent these inequalities is a key skill in algebra and is essential for solving more advanced mathematical problems.

H2: Inequality 4: $-\frac{13 x}{18}+\frac{5}{9} < -\frac{1}{6}$

Finally, let's tackle the fourth inequality: $-\frac{13 x}{18}+\frac{5}{9} < -\frac{1}{6}$. Our objective remains consistent: to isolate x and determine the range of values that satisfy the inequality. This involves algebraic manipulations to simplify the expression and ultimately solve for x. The first step is to eliminate the fractions. We need to find the least common multiple (LCM) of the denominators 18, 9, and 6. The LCM is 18. Multiplying every term in the inequality by 18 will clear the fractions. This gives us: $18(-\frac{13 x}{18}) + 18(\frac{5}{9}) < 18(-\frac{1}{6})$. Simplifying, we get: $-13x + 10 < -3$. Next, we want to isolate the term with x. We subtract 10 from both sides of the inequality: $-13x + 10 - 10 < -3 - 10$, which simplifies to $-13x < -13$. It's crucial to remember that when we divide both sides of an inequality by a negative number, we must flip the inequality sign. In this case, we need to divide both sides by -13 to get x by itself. So, we have: $\frac{-13x}{-13} > \frac{-13}{-13}$, which simplifies to $x > 1$. This solution tells us that x can be any value greater than 1. On a number line, this is represented by an open circle at 1, indicating that 1 is not included in the solution, and an arrow extending to the right, indicating that all values greater than 1 are solutions. The use of an open circle is vital to correctly represent the solution set. This comprehensive step-by-step solution highlights the importance of accurate algebraic manipulation and proper interpretation of inequality symbols.

H2: Representing Solutions on a Number Line

After solving each inequality, the next critical step is to represent the solution set on a number line. The number line provides a visual representation of all possible values of x that satisfy the inequality. This visual aid is immensely helpful in understanding the range of solutions. For inequalities involving 'greater than' (>) or 'less than' (<), we use an open circle on the number line to indicate that the endpoint is not included in the solution. This signifies that values infinitely close to the endpoint are part of the solution, but the endpoint itself is not. Conversely, for inequalities involving 'greater than or equal to' (≥) or 'less than or equal to' (≤), we use a closed circle (or a solid dot) to indicate that the endpoint is included in the solution. The direction of the arrow extending from the circle indicates the range of values that satisfy the inequality. An arrow extending to the right signifies that all values greater than the endpoint are solutions, while an arrow extending to the left signifies that all values less than the endpoint are solutions. For example, if the solution to an inequality is x > 3, we would draw a number line with an open circle at 3 and an arrow extending to the right. If the solution is x ≤ -2, we would draw a number line with a closed circle at -2 and an arrow extending to the left. The number line is a powerful tool for visualizing the solution sets of inequalities, making it easier to grasp the concept of a range of values satisfying a condition rather than just a single value, as in the case of equations. Mastering the representation of solutions on a number line is crucial for a comprehensive understanding of inequalities.

H2: Importance of Understanding Inequalities

Understanding inequalities is paramount in mathematics and its various applications. Inequalities are used to describe situations where values are not necessarily equal but have a specific relationship, such as one value being greater than another. This is extremely common in real-world scenarios. For instance, in economics, inequalities can be used to model budget constraints, where spending must be less than or equal to income. In physics, they can describe the range of possible values for physical quantities, such as the velocity of an object. In computer science, inequalities are used in algorithms to define conditions for loops and conditional statements. Moreover, inequalities form the basis for more advanced mathematical concepts such as linear programming, which is used in optimization problems across various industries, including logistics, finance, and manufacturing. The ability to solve and interpret inequalities is also crucial for understanding calculus, where concepts like limits and continuity rely heavily on inequalities. Inequalities are not just abstract mathematical concepts; they are powerful tools for modeling and solving real-world problems. The skills developed in solving inequalities, such as algebraic manipulation, logical reasoning, and visual representation, are transferable to many other areas of study and professional life. Therefore, a solid understanding of inequalities is an essential building block for anyone pursuing careers in STEM fields, business, or any field that involves quantitative analysis.

H2: Conclusion

In conclusion, solving inequalities and representing their solutions on a number line is a fundamental skill in mathematics. This article has walked through the process step-by-step, from simplifying inequalities to isolating the variable and accurately depicting the solution set on a number line. Each example has highlighted the importance of algebraic manipulation, such as clearing fractions, combining like terms, and remembering to flip the inequality sign when multiplying or dividing by a negative number. The use of open and closed circles on the number line to represent whether the endpoint is included in the solution is a critical detail that must be carefully considered. Furthermore, we have emphasized the broader significance of understanding inequalities. Inequalities are not merely abstract mathematical concepts; they are essential tools for modeling and solving real-world problems across various disciplines. From economics to physics to computer science, inequalities provide a means to describe relationships between quantities that are not necessarily equal, making them invaluable in optimization problems, constraint analysis, and algorithm design. Mastering the art of solving and representing inequalities provides a solid foundation for more advanced mathematical studies and equips individuals with the analytical skills necessary for success in a wide range of fields. Therefore, consistent practice and a thorough understanding of the concepts presented here are key to building mathematical proficiency and problem-solving abilities.