Problem 3 Use interval notation to describ... [FREE SOLUTION]

91影视

Elementary Algebra

David Arnold

$Math Studyset 91影视 Explanations$ Math

2020 Edition

Chapter 2: Problem 3

Use interval notation to describe the solution of: $x-7<-8$.

Short Answer

Expert verified

The solution in interval notation is $(-\infty, -1)$.

Step by step solution

Isolate x on One Side

Start with the inequality: $x - 7 < -8$. The objective is to isolate $x$ by first undoing the subtraction of 7. Add 7 to both sides of the inequality to eliminate it: $x - 7 + 7 < -8 + 7$.

Simplify the Inequality

Simplifying both sides of the inequality gives: $x < -1$. This indicates that $x$ is any value that is less than -1.

Write in Interval Notation

Convert the inequality $x < -1$ into interval notation. Since $x$ can be any value less than -1, it means the interval is open at -1 and extends to negative infinity. Thus, the interval notation is: $(-\infty, -1)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation

Interval notation is a way of writing subsets of real numbers and is commonly used to describe solutions to inequalities. In the case of the inequality $x < -1$, interval notation helps to succinctly state which values $x$ can take. In this scenario, $x$ is any value less than $-1$, which means it spans from $-\infty$ to $-1$, not including $-1$ itself.

Here's how interval notation expresses this:

Open interval: Uses parentheses $(\text{blah}$) to indicate that the endpoint is not included. Hence, for $x < -1$, we have $(-\infty, -1)$.
Infinity: Always uses an open interval $(-\infty )$. Infinity is a concept, not a specific number, and thus can't be "reached" or included.

Using interval notation makes it easier to deal with and communicate the range of numbers involved in inequality solutions.

Isolating Variables

Isolating the variable is a crucial step in solving inequalities, as it simplifies the expression and reveals the values that satisfy the inequality. In the exercise where we had $x - 7 < -8$, the goal was to solve for $x$, making it the subject of the inequality.

To isolate $x$, you can follow these steps:

Identify terms: Recognize the terms on either side of the inequality that complicate $x$'s isolation.
Undo operations: Use inverse operations to remove them. In our example, we added 7 to both sides to counteract the $-7$.

After performing these operations, you will have a much clearer inequality like $x < -1$, which directly describes the condition that $x$ must satisfy.

Maintaining balance by performing the same operations on both sides of the inequality is essential to preserve the truth of the inequality.

Inequality Simplification

Simplifying inequalities is similar to simplifying equations, but it involves careful consideration of the inequality sign. Simplification makes inequalities easier to understand and solve.

In our given problem, after isolating $x$, the inequality $x - 7 < -8$ was simplified to $x < -1$. This simplification involved:

Eliminating terms: Remove terms that obscure the variable. While isolating $x$, unnecessary constants are removed via operation reversal.
Combining like terms: If there were like terms, they would be combined or reduced for clarity. However, $x < -1$ is already as simple as it gets.

Simplifying also helps in ensuring that calculations are correct before proceeding to write the answer in interval notation. You have a clearer statement of the problem that directly leads to its interval representation.

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91影视

Use interval notation to describe the solution of: \(x-7<-8\).

Short Answer

Step by step solution

Isolate x on One Side

Simplify the Inequality

Write in Interval Notation

Key Concepts

Interval Notation

Isolating Variables

Inequality Simplification

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