Problem 97 - Algebra and Trigonometry

Question

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. \(y=1-(x+3)+2 x\) and \(y\) is at least 4

Step-by-Step Solution

Answer

The interval for \(x\) that satisfies the condition is \([6,+\infty)\).

Step 1: Set the y-value to 4

Given the equation \(y = 1-(x+3)+2x\) and \(y \geq 4\), we will first set \(y\) equal to 4 to find the boundary of the interval for \(x\). This leads to the new equation \(4 = 1-(x+3)+2x\).

Step 2: Simplify the function

Rearrange and simplify the function to get \(x\). Simplify the equation to: \(4 = 1 - x - 3 + 2x\) which simplifies further to: \(4 = 1 - 3 + x \), finally simplifying to \(x = 4 - 1 + 3 = 6\). Thus \(x = 6\) is the boundary of the interval.

Step 3: Construct the interval notation

The interval notation can now be constructed. Since \(y\) is 'at least' 4, the interval of \(x\) that satisfies this will be \([6,\infty)\). The reason being that for all values of \(x\) greater than or equal to 6, \(y\) will be 4 or more.

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements that relate expressions with terms like 'greater than' (\t>), 'less than' (\t<), 'greater than or equal to' (\t\geq), or 'less than or equal to' (\t\leq). Unlike equations, which state that two expressions are identical, inequalities express a range of possible solutions. In educational terms, comprehending inequalities is akin to learning the language of ranges in mathematics.

For instance, if we have an inequality such as \t\(y \geq 4\), it is saying that \t\(y\) is greater than or equal to 4. This can represent many possible values of \t\(y\), and it's our job to find all values of \t\(x\) that will make this inequality true. A crucial part of solving these is expressing the solution set, and interval notation succinctly signifies the range of values that satisfy the inequality. In our exercise, the interval \t\([6,\infty)\) means all values from 6 to positive infinity, including 6 itself, satisfy the given conditions.

Solving Algebraic Equations

Solving algebraic equations is a cornerstone of mathematics. An equation is a statement that asserts the equivalence of two expressions, often incorporating variables like \t\(x\) and \t\(y\), which stand in for unknown values we're solving for. In the context of education, mastering these equations enables learners to move from the known to the unknown, translating complex problems into simpler, operable terms.

Steps to Solve an Equation

First, assess what you're given and what you're solving for. In our exercise, we needed to find the value of \t\(x\) that satisfies the equation \t\(y=1-(x+3)+2x\) when \t\(y = 4\).
Next, rearrange the terms: combine like terms and isolate the variable on one side of the equation.
Finally, calculate the value of the variable. In this case, \t\(x=6\).
The skill of solving algebraic equations is fundamental not only in algebra but also across various fields of mathematics and science.

The Role of Trigonometry

Trigonometry is yet another fundamental branch of mathematics, albeit not directly showcased in our exercise related to inequalities and algebra. This field specializes in the study of triangles, specifically the relationships between their angles and sides. In an educational setup, trigonometry stands as a building block for higher mathematics and many real-world applications, including engineering, physics, and geography.

It provides tools like sine, cosine, and tangent, which are ratios of the sides of right-angled triangles that help relate the angles to the lengths of the sides. Trigonometry is crucial when dealing with periodic phenomena like sound waves, light waves, and even the cyclical nature of the stock market. While it wasn't explicitly used in the provided exercise, trigonometry's principles underpin much of the calculations involved in more sophisticated areas of algebra and calculus.