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Chapter 8: Problem 72

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ x \leq 6-\frac{1}{2} x \text { and } \frac{1}{2} x+1 \geq 3 $$

Short Answer

Expert verified
The solution set is \( [4, 4] \), and it represents just the point \( x = 4 \).

Step by step solution

01

Solve the First Inequality

The first inequality given is \( x \leq 6 - \frac{1}{2}x \). To solve for \( x \), first add \( \frac{1}{2}x \) to both sides:\[ x + \frac{1}{2}x \leq 6 \]This simplifies to:\[ \frac{3}{2}x \leq 6 \]Next, divide both sides by \( \frac{3}{2} \) to isolate \( x \):\[ x \leq \frac{6}{\frac{3}{2}} = 6 \times \frac{2}{3} = 4 \]Thus, the solution to the first inequality is \( x \leq 4 \).
02

Solve the Second Inequality

The second inequality is \( \frac{1}{2}x + 1 \geq 3 \). Begin by subtracting 1 from both sides:\[ \frac{1}{2}x \geq 2 \]Now, multiply both sides by 2 to solve for \( x \):\[ x \geq 4 \]Thus, the solution to the second inequality is \( x \geq 4 \).
03

Combine the Solutions

We need to find values of \( x \) that satisfy both inequalities simultaneously. From Step 1, we have \( x \leq 4 \), and from Step 2, we have \( x \geq 4 \). The overlap or intersection of these solutions is \( x = 4 \).
04

Graph the Solution

On a number line, draw a closed dot on \( x = 4 \) to indicate that 4 is included. There are no other values satisfying both inequalities, so the graph is just a single point, \( x = 4 \).
05

Write the Solution in Interval Notation

The solution is just the number 4 itself. In interval notation, this is represented as \([4, 4]\), indicating the single point at 4.

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Key Concepts

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

Solving Inequalities
Inequalities are mathematical expressions that involve a comparison between two values using symbols like \(<, \leq, >, \geq\). Solving inequalities means finding the set of values that makes the inequality true. Here's a friendly guide on how to tackle compound inequalities like the ones in our exercise.

  • Step-by-Step Approach: Break down the compound inequality into two separate inequalities. In our problem, we have:
    • \( x \leq 6 - \frac{1}{2}x \)
    • \( \frac{1}{2}x + 1 \geq 3 \)
  • Simplifying Each Inequality: For the first inequality, add \( \frac{1}{2}x \) to both sides to isolate \( x \), and for the second inequality, simplify by solving for \( x \).
  • Combining Results: Once each inequality is solved, merge the solutions to find the common values. This gives you the values that satisfy the entire compound inequality.
Remember, inequalities are like equations, but with more flexibility in their solutions since they describe a range of possible solutions rather than just one number.
Interval Notation
Interval notation is a shorthand way of writing solutions to inequalities. It's particularly useful because it allows you to illustrate the range of values that your solution covers.

  • Understanding the Symbols: In interval notation, square brackets \([ ]\) are used when the endpoints are included, indicating 'closed' intervals. Parentheses \(( )\) are used when they aren't included, indicating 'open' intervals.
  • Applying this to the Exercise: For our compound inequality, the solution after combining is just one point: \( x = 4 \). This means the interval notation is a closed interval, \([4, 4]\).
  • Benefits of Interval Notation: It provides an easy-to-read structure that communicates exactly which numbers are part of the solution. It helps quickly convey complex mathematical concepts.
While reading or writing interval notation, always remember to observe whether the endpoints are included or not, which hinges on the inequality symbols used.
Graphing Solutions
Graphing the solutions of an inequality on a number line helps visualize the set of possible answers. It's a simple way to ensure your solution makes intuitive sense.

  • Using a Number Line: Place a number line on your paper, and use dots or shading to indicate the solution set. A filled-in circle shows inclusion of the endpoint, while an open circle indicates it鈥檚 not included.
  • Our Graph for \( x = 4 \): Since our solution set \( x = 4 \) includes the number 4, we place a solid dot on the number line at \( x = 4 \). This visually clarifies to anyone that this is the exact solution point.
  • Importance of Graphing: Graphing not only aids visual learners but also reinforces confidence in the solution by providing an immediate check if it aligns with the mathematical solution.
Graphing can be very empowering for students, leveraging visuals to make abstract concepts more concrete and easier to grasp.

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Most popular questions from this chapter

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{4}{m^{2}-9}+\frac{5}{m^{2}-m-12}=\frac{7}{m^{2}-7 m+12}$$

Graph each function. See Objective 5. $$ g(x)=-0.25 x $$

Perform the operations and simplify. $$\frac{2 x^{2}-2 x-4}{x^{2}+2 x-8} \cdot \frac{3 x^{2}+15 x}{x+1} \div \frac{100-4 x^{2}}{x^{2}-x-20}$$

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{2}{x-1}-\frac{2 x}{x^{2}-1}-\frac{x}{x^{2}+2 x+1}$$

Perform the operations and simplify. $$\frac{\frac{h}{h^{2}+3 h+2}}{\frac{4}{h+2}-\frac{4}{h+1}}$$
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