Mastering Inequalities And Expressions A Comprehensive Guide

This article aims to provide a detailed exploration of solving inequalities and simplifying expressions, fundamental concepts in mathematics. We will delve into various techniques and strategies to tackle these problems effectively. Understanding these concepts is crucial for students and anyone involved in mathematical problem-solving. Let's embark on this journey to unravel the intricacies of inequalities and expressions.

1. Solving the Inequality: $2x - 1.5 > 5x - 6$

When solving inequalities, the primary goal is to isolate the variable on one side of the inequality sign. This involves performing operations on both sides of the inequality while maintaining its balance. The key is to remember that multiplying or dividing by a negative number reverses the direction of the inequality sign. In this section, we will walk through the steps to solve the inequality $2x - 1.5 > 5x - 6$, highlighting the underlying principles and techniques.

Our first step in solving this inequality is to gather all the terms containing the variable $x$ on one side and the constant terms on the other side. To achieve this, we can subtract $2x$ from both sides of the inequality. This yields: $-1.5 > 3x - 6$. Next, we want to isolate the term with $x$. To do this, we add 6 to both sides, resulting in: $4.5 > 3x$. Now, we need to isolate $x$ completely. We can accomplish this by dividing both sides of the inequality by 3. This gives us: $1.5 > x$, which can also be written as $x < 1.5$. This final form represents the solution to the inequality.

Therefore, the solution to the inequality $2x - 1.5 > 5x - 6$ is $x < 1.5$. This means that any value of $x$ that is less than 1.5 will satisfy the original inequality. To better understand this, we can visualize this solution on a number line. We would represent all values less than 1.5, using an open circle at 1.5 to indicate that 1.5 itself is not included in the solution set. The solution set includes all numbers to the left of 1.5 on the number line. This graphical representation aids in the comprehension of the solution.

It's also crucial to emphasize the importance of checking the solution. To verify the accuracy of our solution, we can substitute a value less than 1.5 into the original inequality and check if it holds true. For example, let's substitute $x = 1$ into the original inequality: $2(1) - 1.5 > 5(1) - 6$. This simplifies to $0.5 > -1$, which is a true statement. This confirms that our solution, $x < 1.5$, is correct. By consistently applying these steps and checking the solution, we can confidently solve various linear inequalities.

2. Evaluating: rac{1}{3} - 1.2x - rac{1}{2}

Evaluating expressions often involves simplifying and combining like terms. This requires a solid understanding of arithmetic operations, including fractions and decimals. In this section, we will evaluate the expression rac{1}{3} - 1.2x - rac{1}{2}, focusing on the techniques for handling fractions and combining them with decimal terms. This process will enhance your ability to work with algebraic expressions and numeric values effectively.

The first step in evaluating the expression rac{1}{3} - 1.2x - rac{1}{2} is to focus on the constant terms, which are the fractions rac{1}{3} and -rac{1}{2}. To combine these fractions, we need to find a common denominator. The least common denominator (LCD) for 3 and 2 is 6. We convert each fraction to an equivalent fraction with a denominator of 6. Thus, rac{1}{3} becomes rac{2}{6} and -rac{1}{2} becomes -rac{3}{6}. Now, we can rewrite the expression as rac{2}{6} - 1.2x - rac{3}{6}. Combining the fractions, we get rac{2}{6} - rac{3}{6} = -rac{1}{6}.

Now, the expression is simplified to -rac{1}{6} - 1.2x. The term $-1.2x$ contains the variable $x$, and we cannot combine it directly with the constant term -rac{1}{6}. To proceed further, we can convert the fraction -rac{1}{6} to a decimal. Dividing 1 by 6, we get approximately $-0.1667$. Therefore, the expression can be rewritten as $-0.1667 - 1.2x$. However, the expression is already in its simplest form, as we cannot combine the constant term with the term containing the variable $x$.

Therefore, the simplified form of the expression rac{1}{3} - 1.2x - rac{1}{2} is -rac{1}{6} - 1.2x, or equivalently, $-0.1667 - 1.2x$. The final form depends on whether you prefer to keep the constant term as a fraction or convert it to a decimal. The key is to perform the arithmetic operations accurately and to combine like terms effectively. Understanding how to manipulate fractions and decimals is crucial in various mathematical contexts, making this evaluation a fundamental skill.

3. Solving the Inequality: rac{3}{4}(x + 15) + 1 \leq (x - 2) + 5

Solving inequalities that involve fractions and parentheses requires careful application of the distributive property and order of operations. In this section, we will address the inequality rac{3}{4}(x + 15) + 1 ext{less than or equal to} (x - 2) + 5 by systematically breaking it down into manageable steps. This includes distributing, combining like terms, and isolating the variable. A methodical approach is essential for achieving the correct solution.

Our initial step in solving the inequality rac{3}{4}(x + 15) + 1 ext{less than or equal to} (x - 2) + 5 is to eliminate the parentheses. This involves applying the distributive property, which states that $a(b + c) = ab + ac$. On the left side, we distribute rac{3}{4} across $(x + 15)$: rac{3}{4} * x + rac{3}{4} * 15 = rac{3}{4}x + rac{45}{4}. So, the left side becomes rac{3}{4}x + rac{45}{4} + 1. On the right side, we have $(x - 2) + 5$, which simplifies to $x - 2 + 5 = x + 3$. Therefore, the inequality is now rac{3}{4}x + rac{45}{4} + 1 ext{less than or equal to} x + 3.

Next, we simplify both sides of the inequality by combining like terms. On the left side, we need to combine the constants rac{45}{4} and $1$. To do this, we express 1 as a fraction with a denominator of 4: 1 = rac{4}{4}. So, rac{45}{4} + rac{4}{4} = rac{49}{4}. The left side now becomes rac{3}{4}x + rac{49}{4}. The right side remains $x + 3$. The inequality is now rac{3}{4}x + rac{49}{4} ext{less than or equal to} x + 3. To eliminate fractions, we can multiply both sides of the inequality by 4. This gives us 4 * (rac{3}{4}x + rac{49}{4}) ext{less than or equal to} 4 * (x + 3), which simplifies to $3x + 49 ext{less than or equal to} 4x + 12$.

We now need to isolate the variable $x$. Subtracting $3x$ from both sides gives $49 ext{less than or equal to} x + 12$. Next, subtract 12 from both sides to isolate $x$: $49 - 12 ext{less than or equal to} x$, which simplifies to $37 ext{less than or equal to} x$. This means that $x$ is greater than or equal to 37. The solution to the inequality is $x ext{greater than or equal to} 37$. It is important to check this solution by substituting a value greater than or equal to 37 into the original inequality to ensure it holds true. By systematically applying these steps, we can solve complex inequalities with confidence.

4. Simplify the Expression:

Simplifying expressions is a crucial skill in algebra, enabling us to rewrite complex algebraic expressions into a more manageable form. This often involves combining like terms, applying the distributive property, and following the order of operations. In this section, we will explore various simplification techniques to enhance your ability to work with algebraic expressions. The goal is to break down complex expressions into their simplest form, making them easier to understand and manipulate.

To effectively simplify an expression, it is essential to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the expression $3x + 2y - 5x + 7y$, the terms $3x$ and $-5x$ are like terms, and the terms $2y$ and $7y$ are like terms. We can combine $3x$ and $-5x$ to get $-2x$, and we can combine $2y$ and $7y$ to get $9y$. Therefore, the simplified expression is $-2x + 9y$. This process of combining like terms is fundamental in simplifying algebraic expressions and making them more concise.

The distributive property is another essential tool in simplifying expressions. This property allows us to multiply a single term by each term inside a set of parentheses. For example, in the expression $4(2x - 3)$, we apply the distributive property by multiplying 4 by each term inside the parentheses: $4 * 2x - 4 * 3 = 8x - 12$. This simplifies the expression by removing the parentheses. The distributive property is particularly useful when dealing with expressions that involve parentheses and multiple terms, enabling us to expand and simplify them effectively. Applying the distributive property correctly is a critical step in simplifying algebraic expressions.

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for simplifying expressions correctly. This order dictates the sequence in which operations should be performed. First, we address any expressions within parentheses. Then, we evaluate exponents. Next, we perform multiplication and division from left to right. Finally, we carry out addition and subtraction from left to right. Following the order of operations ensures that we arrive at the correct simplified form of an expression. For example, in the expression $2 + 3 * 4$, we first perform the multiplication: $3 * 4 = 12$, and then the addition: $2 + 12 = 14$. Thus, the simplified expression is 14. Adhering to the order of operations is vital for accurate simplification.

In conclusion, simplifying expressions involves a combination of techniques, including combining like terms, applying the distributive property, and following the order of operations. By mastering these skills, you can effectively simplify complex algebraic expressions into more manageable forms. This not only makes the expressions easier to understand but also facilitates further mathematical manipulations and problem-solving. Regular practice and a solid understanding of these principles will enhance your ability to simplify expressions with confidence and accuracy.