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Chapter 6: Problem 16

Solve the inequality and graph its solution. \(-2 b \leq-8\)

Short Answer

Expert verified
The solution to the inequality is \(b \geq 4\).

Step by step solution

01

Understand and Rewrite the Inequality

The given inequality is \(-2b \leq-8\). This inequality means \(-2b\) is less than or equals to \(-8\). Our goal is to isolate \(b\). In order to do this, it is necessary to divide by \(-2\).
02

Divide by Negative Coefficient

Divide each side of the inequality by \(-2\) to isolate \(b\). Here we must remember an important rule: When you multiply or divide an inequality by a negative number, you must reverse the inequality symbol. Hence we get, \(b \geq 4\).
03

Represent the Solution Graphically

To graph this on a number line, place a filled circle on \(4\) (because \(4\) is included in the solution) and draw a line extending to the right from \(4\) (since it includes all numbers greater than \(4\)).

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Key Concepts

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

Graphing Inequalities
Visualizing inequalities on a graph can make them more understandable. To graph the inequality solution, you need two parts: the value where the inequality becomes true and the direction of the graph. For instance, with the inequality \(b \geq 4\), we start at \(b = 4\). Here, the value \(4\) itself can be part of the solution set, so we use a filled circle.
  • Draw a filled circle on the number line at 4.
  • Extend a line to the right, showing all values greater than 4.

These steps represent all possible solutions, helping you to quickly understand the range of numbers that satisfy the inequality.
Solving Inequalities
Solving inequalities is like solving equations, but with a few more twists. You start by isolating the variable, focusing on operations like addition, subtraction, multiplication, or division.
For the inequality \(-2b \leq -8\), the goal is to find what \(b\) can be. To isolate \(b\), you divide both sides by \(-2\). This gives you:
\[b \geq 4\]
Once the inequality sign is adjusted correctly, you can efficiently solve and find all possible values for the variable.
Reversing Inequality Signs
A key detail when working with inequalities is what happens when you multiply or divide by a negative number. If you need to do this, you must reverse the inequality sign to maintain a true statement.
In our example, the original inequality is \(-2b \leq -8\). Dividing by \(-2\) means we switch from \(\leq\) to \(\geq\), resulting in \(b \geq 4\).
  • Always reverse the sign with negative division or multiplication.
  • This is a crucial step to remember, as skipping it can lead to incorrect solutions.

Paying attention to this rule ensures that your inequalities remain valid and accurate.
Number Line Representation
Number lines offer a simple way to express the solution set of an inequality. They provide a visual depiction that complements the algebraic form.
Let's take \(b \geq 4\) as our final inequality. On a number line:
  • Mark a filled circle at 4, since 4 is included.
  • Draw a line extending to the right, showing that all greater numbers are solutions.

Number line diagrams make it easier to intuitively grasp the span of solutions that fit the inequality, by providing a clear and immediate picture of which numbers satisfy the condition.

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