Problem 11 - Elementary Algebra

Question

Find the values of \(x\) that satisfy the given continued inequality. $$ 4

Step-by-Step Solution

Answer

Question: Solve the inequality \(4 < x - 5 < 12\) and determine the values of \(x\) that satisfy the given continued inequality. Answer: The values of \(x\) that satisfy the given continued inequality are those between 9 and 17, exclusive, represented as \(9 < x < 17\).

Step 1: Rewrite the inequalities separately

Rewrite the given continued inequality into the two separate inequalities: 1. \(4 < x-5\) 2. \(x-5 < 12\)

Step 2: Solve the first inequality for x

To solve \(4 < x-5\), add 5 to both sides of the inequality: $$ 4+5 < x-5+5 $$ This simplifies to: $$ 9 < x $$

Step 3: Solve the second inequality for x

To solve \(x-5 < 12\), add 5 to both sides of the inequality: $$ x-5+5 < 12+5 $$ This simplifies to: $$ x < 17 $$

Step 4: Analyze the solutions

We have the following individual inequalities: 1. \(9 < x\) 2. \(x < 17\) To find the values of \(x\) that satisfy the given continued inequality, we need to find the intersection of the solutions of these two inequalities. Since both inequalities are true at the same time, we can combine them as a single inequality: $$ 9 < x < 17 $$ This means the values of \(x\) that satisfy the given continued inequality are those between 9 and 17, exclusive.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It's like a puzzle where you solve for unknowns represented by variables such as \(x\) or \(y\). One of algebra’s main goals is to solve equations and inequalities by finding the values that make those statements true. This involves a variety of operations like addition, subtraction, multiplication, and division.In algebra, understanding how to manipulate and balance equations is key. This process ensures that any operations performed on one side of the equation are also done to the other side. This keeps the equation balanced. Algebra often involves expressions, which are combinations of numbers and variables connected by operations, and equations, which have an equality sign that separates two expressions.

By mastering algebra, you gain the tools to handle more complex mathematical concepts and everyday problems.

Linear Inequalities

Linear inequalities are very similar to linear equations, but instead of having an equal sign, they have inequality symbols. These can include greater than (\(>\)), less than (\(<\)), greater than or equal to (\(\geq\)), and less than or equal to (\(\leq\)).With linear inequalities, you often find a range of values that satisfy the inequality rather than a single solution. For instance, in the inequality \(x < 17\), any value less than 17 will satisfy the inequality.

One characteristic of linear inequalities is that when graphing them, they often appear as a shaded region on one side of a boundary line. This line represents the equality part, like in \(x = 17\), and the shading indicates all the numbers that fulfill the inequality condition.

• When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality symbol.
• This is a crucial rule to remember while solving inequalities.

By solving linear inequalities, you can determine entire sets of solutions that apply to real-world scenarios.

Solving Inequalities

Solving inequalities is an extension of solving equations. The key difference is the type of symbol used. The process involves similar steps to solving equations, yet with an additional consideration for the inequality symbol.The first step is to simplify both sides of the inequality if needed. After that, like in the example \(4 < x-5 < 12\), you may need to separate the compound inequality into individual inequalities to solve them more easily. In the sample inequality, we solved separately:

• \(4 < x-5\) becomes \(9 < x\) after adding 5 to both sides.
• \(x-5 < 12\) turns into \(x < 17\) by adding 5 to both sides too.

After solving each inequality separately, you can combine the results to find the solution set that fulfills both conditions. In our example, it resulted in \(9 < x < 17\), meaning \(x\) can be any value between 9 and 17, exclusive.Always remember to:

• Check if the inequality needs reversing, especially when multiplying or dividing by negatives.
• Express the final answer clearly, which often involves writing the values as a range or interval.

Through this approach, solving inequalities becomes straightforward and directly applicable to many real-life contexts.