Problem 52 Solve and write the answer in se... [FREE SOLUTION]

Chapter 14: Problem 52

Solve and write the answer in set-builder notation. Graph the solution set. $$-4 x<8$$

Short Answer

Expert verified

The solution to the inequality is $x > -2$. In set-builder notation, it is written as $ \{x | x \in R, x > -2 \}$. The graphical representation can be shown on a number line with an open point at -2 and an arrow pointing towards values greater than -2.

Step by step solution

Solving the Inequality

To isolate $ x $, divide both sides of the equation by $-4$. Remember when dividing an inequality by a negative number, we must reverse the direction of the inequality to keep it valid. Therefore, $-4x < 8$ becomes $x > -2$.

Conversion into Set-builder Notation

Set builder notation is a method of specifying a set of elements that satisfy a certain condition. In this case, the condition is $x > -2$. So, the solution in set-builder notation is $ \{x | x \in R, x > -2 \} $. This indicates that the solution set includes all real numbers $x$ that are greater than -2.

Graphical Representation of the Solution

Using a number line, a point is plotted at -2 and since the inequality is $x > -2$ (not $x \geq -2$) an open circle is drawn at -2. Then, as $x$ includes the numbers greater than -2, an arrow is drawn to right indicating all numbers greater than -2 are included in the solution.

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Key Concepts

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

Understanding Set-Builder Notation

Set-builder notation is a concise way to describe a set of numbers by stating a property that the members of the set must satisfy. In the context of inequalities, it allows us to clearly express the solution set of an inequality.
For example, when we want to express all real numbers that are greater than $-2$, we write it as $ \{x | x \in \mathbb{R}, x > -2\} $.
This notation means "the set of all $x$ such that $ x$ belongs to the real numbers, and $x$ is greater than $-2$."
Set-builder notation helps to avoid lengthy descriptions, making it easier to quickly identify the characteristics of the set you are dealing with.

Solving Inequalities

Solving inequalities involves finding all values of the variable that make the inequality true. Unlike equations that have a single or finite number of solutions, inequalities often have an infinite number of solutions.
Here are some key steps when solving an inequality:

Perform the same operation on both sides to maintain balance.
If you multiply or divide by a negative number, flip the inequality sign.

For example, to solve $-4x < 8$, we would divide by $-4$, and flip the sign:
1. Divide both sides by $-4$: $ x > -2 $
This step ensures we have a clearer understanding of how to manage inequalities and helps confirm the solution is accurate.

Graphing Inequalities

Graphing inequalities on a number line provides a visual representation of the solution, which can make understanding the range of values easier.
To graph an inequality like $x > -2$:

First, locate $-2$ on the number line.
Since the inequality sign is $>$ and not $\geq$, this means it doesn't include $-2$, so we use an open circle.
Draw a line or arrow starting from this point, extending to the right because $x$ is greater than $-2$.

This method helps to visually understand the sets of solutions and how they relate to the number line.

Number Line Representation

A number line is a powerful way to display the solutions of inequalities. It allows us to quickly observe where a variable can lie in relation to other numbers.
With inequalities like $x > -2$, the number line is marked with an open circle at $-2$, indicating $-2$ is not included in the solution.
An arrow is then drawn to the right of $-2$, covering all numbers greater than $-2$.
Number lines are helpful tools for seeing solutions in a straightforward manner, aiding skills in comprehension and comparison of numbers across the line.

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91影视

Solve and write the answer in set-builder notation. Graph the solution set. $$-4 x<8$$

Short Answer

Step by step solution

Solving the Inequality

Conversion into Set-builder Notation

Graphical Representation of the Solution

Key Concepts

Understanding Set-Builder Notation

Solving Inequalities

Graphing Inequalities

Number Line Representation

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