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Chapter 12: Problem 37

Solve each inequality. Then graph the solution on a number line. $$2 a-3 \geq 9$$

Short Answer

Expert verified
The solution is \(a \geq 6\) and is graphed with a closed circle on 6, shading to the right.

Step by step solution

01

Isolate the Variable Term

Start with the given inequality: \[2a - 3 \geq 9\]To isolate the term with the variable, add 3 to both sides of the inequality:\[2a - 3 + 3 \geq 9 + 3\]Simplify both sides of the equation:\[2a \geq 12\]
02

Solve for the Variable

Now, we need to solve for \(a\) by dividing both sides by 2:\[\frac{2a}{2} \geq \frac{12}{2}\]After dividing, we get:\[a \geq 6\]
03

Graph the Solution on a Number Line

To graph \(a \geq 6\) on a number line, draw a number line and place a closed circle on 6 to indicate that 6 is included in the solution. Then, shade the line to the right of 6 to represent all numbers greater than or equal to 6.

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Key Concepts

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

Solving Inequalities
Solving inequalities is quite similar to solving equations, but with a twist. Instead of looking for just a specific value for the variable, you are finding a range of values that satisfy the inequality. Given the inequality \(2a - 3 \geq 9\), the first task is to isolate the variable on one side. To do this, you add or subtract the same amount from both sides. In our example, we add 3 to both sides, simplifying to \(2a \geq 12\). Next, to isolate \(a\), divide every term by 2, resulting in \(a \geq 6\).
  • Always perform the same operation on both sides of the inequality.
  • Remember the golden rule: if you multiply or divide by a negative number, reverse the inequality sign!
This approach ensures that the inequality remains balanced and accurate through your calculations.
Graphing on a Number Line
Graphing solutions to inequalities on a number line helps visually represent the range of possible solutions. After solving for \(a\) in the example, you get \(a \geq 6\). This tells you that any number greater than or equal to 6 is a solution. To graph it:
  • Draw a number line and locate the point for 6.
  • Place a closed circle on 6 to indicate that 6 is included in the solutions.
  • Shade the number line to the right of 6 to illustrate all solutions that are greater than or equal to 6.

The closed circle is significant as it shows that the point itself is part of the solution set. For example, if it were \(a > 6\), you would use an open circle instead.
Prealgebra Concepts
Prealgebra serves as the foundation for all higher-level math you'll encounter. It sets the stage with basic mathematical operations, number properties, and simple equations. Inequalities, like the one we've solved, are an essential part of prealgebra. They use similar steps to those in equation solving but require a keen awareness of the inequality signs and their properties.

Key prealgebra concepts involved in solving inequalities are:
  • Understanding and using number properties (addition, subtraction, multiplication, division).
  • Keeping equations balanced by performing the same operations on either side.
  • Recognizing directionality: knowing when to flip inequality signs based on multiplication or division by negative values.

Mastery in these areas provides a solid groundwork for advancing into algebra, where you will encounter more complex problems and greater mathematical challenges.

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Most popular questions from this chapter

Express each decimal or fraction as a percent. Round to the nearest hundredth percent, if necessary. $$\frac{15}{8}$$

Find the median for each set of data. Round to the nearest tenth, if necessary. $$0.4,1.3,0.8,1.8$$

Order each set of decimals from least to greatest. (page 742 ) $$0.3,1.4,0.6,1.5,0.2,0.8,1.2$$

Order \(6.1 \times 10^{4}, 6100,6.1 \times 10^{-5}, 0.0061,\) and \(6.1 \times 10^{-2}\) from least to greatest.

Find each product. $$\frac{1}{6} \cdot \frac{1}{3}$$ $$\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}$$
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