Problem 15 - Elementary Algebra

Question

Find the values of \(x\) that satisfy the given continued inequality. $$ 9<\frac{-4 x+5}{-2}<14 $$

Step-by-Step Solution

Answer

The values of x that satisfy the given continued inequality are those that are greater than 23/4 and smaller than 33/4.

Step 1: Split the inequality into two parts

First, let's split the given continued inequality into two separate individual inequalities: $$ 9 < \frac{-4 x + 5}{-2} $$ and $$ \frac{-4 x + 5}{-2} < 14 $$ We will solve both inequalities separately and then analyze the common solutions.

Step 2: Solve the first inequality

Let's solve the first inequality: $$ 9 < \frac{-4 x + 5}{-2} $$ To get rid of the fraction, we'll multiply both sides of the inequality by \(-2\). Since we are multiplying an inequality by a negative number, we need to change the direction of the inequality sign. It becomes: $$ -18 > -4x + 5 $$ Now, subtract 5 from both sides: $$ -23 > -4x $$ Divide both sides by \(-4\), and remember to change the direction of the inequality sign: $$ x > \frac{23}{4} $$

Step 3: Solve the second inequality

Now let's solve the second inequality: $$ \frac{-4 x + 5}{-2} < 14 $$ Similar to the first inequality, we'll multiply both sides by \(-2\) and reverse the inequality sign: $$ -28 > -4x + 5 $$ Subtract 5 from both sides: $$ -33 > -4x $$ Divide both sides by \(-4\), and remember to change the direction of the inequality sign: $$ x < \frac{33}{4} $$

Step 4: Find the common solutions

Now, we need to find the common values of \(x\) that satisfy both inequalities. Based on the solutions we found, we can write the combined inequality as: $$ \frac{23}{4} < x < \frac{33}{4} $$ So the values of \(x\) that satisfy the given continued inequality are those that are greater than \(\frac{23}{4}\) and smaller than \(\frac{33}{4}\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a critical skill in solving inequalities, involving the rearrangement and simplification of expressions to isolate the variable of interest. When solving an inequality like the given problem, breaking it down into simpler pieces can be a wise strategy. Start by focusing on each part separately to manage complexity effectively.

In our specific exercise, we had a continuous inequality which was split into two distinct inequalities. This approach allows you to address each inequality individually, simplifying the process of finding a range of solutions.

A key technique in the manipulation process is dealing with fractions. By aiming to eliminate fractions, we simplify the expressions, making them easier to handle. In this problem, multiplying both sides of the inequality by \(-2\) helped us get rid of the fraction, paving the way to focus solely on linear expressions.

Remember, algebraic manipulation requires careful attention to the operations you perform. Any changes to the inequality, like multiplying or dividing, must be accurately tracked and applied to both sides of the equation.

Inverse Operations

Inverse operations are essential tools in solving mathematic equations and inequalities, involving undoing the effect of an operation. They help isolate variables, leading you to the solution.

In simpler terms, each mathematical operation has an inverse:

• Addition and subtraction are inverses of each other
• Multiplication and division are inverses of each other

By applying inverse operations, we transform complex equations into more manageable forms.

In our current inequality problem, inverse operations played a pivotal role. For instance, after multiplying both sides of an inequality by \(-2\) to eliminate the fraction, addition and subtraction were used to move constant terms. This step-by-step application of inverse operations systematically reduced the complexity of the equations.

Always remember, deploying inverse operations need caution, especially when dealing with inequalities, as certain operations may result in reversing the inequality sign, particularly when using negative numbers.

Inequality Properties

The properties of inequalities are key tools for understanding and manipulating these expressions. Here are the main properties essential for anyone tackling inequalities:

• Transposition: You can add or subtract the same number from both sides of an inequality, preserving the direction of the inequality.
• Multiplication and Division: Multiplying or dividing both sides by a positive number retains the inequality's direction. However, doing so with a negative number requires reversing the inequality sign.
• Splitting Complex Inequalities: This is the method used in continuous inequalities, where separating into simpler inequalities makes solving them feasible.

In our exercise, these properties helped navigate through the complexities of manipulation effectively.

Beginning with separate inequalities allowed us to test solutions individually, finally giving us a viable solution range that satisfies all parts of the inequality.

Always be mindful of the operations involved. Executing them carelessly can lead to incorrect results, notably in reversing inequality signs without necessity. Properly applying these properties ensures accurate and effective solutions to inequality problems.