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Chapter 14: Problem 16

Solve and write the answer in interval notation. $$x+8 \geq-14$$

Short Answer

Expert verified
The solution is \([-22, +\infty)\).

Step by step solution

01

Remove the Addition on the Left Side

In order to isolate x on one side of the inequality, subtract 8 from both sides of the inequality \(x + 8 \geq -14\). This gives \(x \geq -14 -8\).
02

Simplify the Right Side

The subtraction on the right side simplifies to \(x \geq -22\). This means x is greater than or equal to -22.
03

Convert to Interval Notation

The solution \(x \geq -22\) in interval notation is written as \([-22, +\infty)\). The square bracket, [ -22, means including -22, and the round bracket, \(+\infty)\), means not including infinity as infinity is not a real number.

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Key Concepts

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

Interval Notation
When you solve inequalities, you often present your solutions using interval notation. It is a way to display the set of all numbers that satisfy the inequality. Interval notation uses brackets "[ ]" and parentheses "( )" to describe the intervals on a number line.
The brackets "[ ]" represent that the endpoints are included in the interval, while parentheses "( )" indicate that the endpoints are excluded.
  • For instance, \([-22, +\infty)\) tells us that the number \(-22\) is included in the solution, but infinity is not, as infinity is conceptual rather than a fixed point.
  • Another example: The notation \((5, 10]\) means all numbers starting just above 5, up to and including 10.
By mastering interval notation, you can efficiently communicate the range of values that satisfy more complex inequalities.
Inequality Solution
Solving inequalities is similar to solving equations, but with a few extra steps and rules. When you see an inequality like \(x + 8 \geq -14\), the main goal is to solve for \(x\), just like an equation.
However, always be careful of inequality properties, especially when multiplying or dividing by negative numbers, which flips the inequality sign.
  • First, isolate the variable. For \(x + 8 \geq -14\), subtract 8 from both sides, simplifying the inequality to \(x \geq -22\).
  • Next, ensure the direction of the inequality is maintained unless you have multiplied or divided by a negative number.
Overall, paying attention to these steps helps you reach the correct solution, whether you're solving simpler or more complicated inequalities.
Algebraic Expressions
Algebraic expressions are combinations of numbers and variables linked by mathematical operations such as addition, subtraction, multiplication, and division. They don't have an equal sign and can represent patterns or relationships.
For example, in \(x + 8\) from the original exercise, \(x\) represents a variable whose value we aim to find.
  • The term *x* can be any number that, when added to 8, satisfies the inequality condition.
  • Expressions like \(x + 8\) are manipulated to find solutions to inequalities like \(x + 8 \geq -14\).
Understanding algebraic expressions is crucial because they allow you to model and solve real-world problems by expressing constraints or relationships mathematically.

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