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Chapter 3: Problem 23

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ x+6<-14 $$

Short Answer

Expert verified
The solution to the inequality is \(x < -20\).

Step by step solution

01

Understand the Inequality

The inequality given is \(x + 6 < -14\). Our goal is to find the values of \(x\) that satisfy this inequality.
02

Isolate the Variable

To isolate \(x\), subtract 6 from both sides of the inequality. This gives us: \(x + 6 - 6 < -14 - 6\).
03

Simplify the Expression

Simplify the expression obtained in Step 2. This leads to \(x < -20\). Now \(x\) is isolated, and the inequality is solved.

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Key Concepts

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

Understanding Algebra
Algebra is a branch of mathematics that helps us make sense of mathematical relationships using symbols and letters. These symbols stand in for numbers and allow us to express equations and inequalities in a generalized form.
Algebra includes the use of variables to express relationships and solve problems. The variable, often represented by symbols like \(x\) or \(y\), acts as a placeholder for unknown values. This allows us to develop formulas and expressions that can be applied to various situations.
  • Algebraic expressions can represent real-world situations.
  • Equations and inequalities are the backbone of algebra, helping us design and solve problems.
  • Algebra is not just theoretical; it applies to numerous fields such as engineering, economics, and science.

Working with algebra involves transforming and simplifying expressions. These manipulations make it easier to understand or solve problems.
The Process of Solving Inequalities
Solving inequalities is very similar to solving equations, with a few important differences. In the problem \(x + 6 < -14\), we are tasked with finding all possible values for \(x\) that satisfy this condition.
When solving inequalities:
  • Isolate the variable. This involves getting \(x\) on one side of the inequality sign, much like you would solve an equation.
  • Perform the same operation on both sides. Just as you would add or subtract numbers in an equation, you do the same in inequalities to maintain balance.
  • Note that multiplying or dividing by a negative number flips the inequality sign. This is a crucial difference from equations that you must remember.

In essence, inequalities allow us to express a range of solutions rather than a single value, providing more extensive information about what \(x\) can be.
Working with Mathematical Expressions
Mathematical expressions form the foundation of equations and inequalities. They are combinations of numbers, variables, and operations like addition or subtraction.
Mathematical expressions do not have an equal sign, unlike equations or inequalities. They serve as building blocks, not only in algebra but in all areas of mathematics.
  • Expressions can be simplified by combining like terms, such as adding or subtracting similar variables.
  • They are often used to solve equations and inequalities by substituting known values to find unknowns.
  • Understanding how to manipulate expressions is crucial for solving problems efficiently and accurately.

When working with expressions, the goal is often to simplify or transform them to make solving more straightforward. This skill is vital for effectively tackling algebraic problems.

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Most popular questions from this chapter

Solve each equation. $$5(2 x-1)-(3 x+4)=4(x+3)-27$$

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The sum of the measure of an angle and three times its complement is \(202^{\circ}\). Find the measure of the angle.

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