Problem 61 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(-1 \leq \frac{2 x-5}{6} \leq 5\)

Step-by-Step Solution

Answer

The solution set is \[ -\frac{1}{2}, 17.5 \].

Step 1: Understand the Given Inequality

The given compound inequality is \[\begin{equation} -1 \leq \frac{2x-5}{6} \leq 5 \end{equation}\]

Step 2: Break Down the Inequality into Two Parts

Break the compound inequality into two separate inequalities: \[\begin{equation} -1 \leq \frac{2x-5}{6} \end{equation}\] \ and \[\begin{equation} \frac{2x-5}{6} \leq 5 \end{equation}\]

Step 3: Solve the First Inequality

Start with \[\begin{equation} -1 \leq \frac{2x-5}{6} \end{equation}\]. Multiply both sides by 6: \[\begin{equation} -6 \leq 2x-5 \end{equation}\]Add 5 to both sides: \[\begin{equation} -1 \leq 2x \end{equation}\]. Now, divide both sides by 2: \[\begin{equation} -\frac{1}{2} \leq x \end{equation}\]

Step 4: Solve the Second Inequality

Now solve \[\begin{equation} \frac{2x-5}{6} \leq 5 \end{equation}\]. Multiply both sides by 6: \[\begin{equation} 2x-5 \leq 30 \end{equation}\]Add 5 to both sides: \[\begin{equation} 2x \leq 35 \end{equation}\]. Divide both sides by 2: \[\begin{equation} x \leq 17.5 \end{equation}\]

Step 5: Combine the Solutions

Combine the results of both inequalities: \[\begin{equation} -\frac{1}{2} \leq x \leq 17.5 \end{equation}\]

Step 6: Graph the Solution Set

Graph the solution set on a number line: Draw a closed circle at \[\begin{equation} x = -\frac{1}{2} \end{equation}\] and another closed circle at \[\begin{equation} x = 17.5 \end{equation}\], with a shaded line connecting the two points indicating all the values between them including the endpoints.

Step 7: Write the Interval Notation

The interval notation for the solution set is: \[\begin{equation} \left[ -\frac{1}{2}, 17.5 \right] \end{equation}\]

Key Concepts

Understanding Compound Inequalities

Compound inequalities involve two separate inequalities that are combined into one statement. They are often joined by the words 'and' or 'or.' An 'and' compound inequality indicates that both conditions must be true simultaneously. For example, in the inequality \(-1 \leq \frac{2x-5}{6} \leq 5\), both inequalities must be satisfied at the same time.
To solve a compound inequality, you split it into two individual inequalities, solve each one separately, and then combine the results. This ensures that all parts of the compound statement hold true.

In the given exercise, we break the compound inequality into two simpler inequalities and solve them one by one. After that, we combine the results to get the final solution set. This step-by-step approach leads to an accurate and clear solution.

Using Interval Notation

Interval notation is a shorthand way of writing inequalities that describe a range of values. It uses brackets and parentheses to show which part of the inequality is included or excluded.

For example:

• \([a, b]\): Includes both endpoints 'a' and 'b'.
• \((a, b)\): Excludes both endpoints.
• \([a, b)\) or \((a, b]\): Includes one endpoint but excludes the other.

In the context of our exercise, after solving the inequalities, the solution set is written as \([-\frac{1}{2}, 17.5]\), which means 'x' includes both -0.5 and 17.5, as well as every number in between. This concise representation makes it easier to understand and communicate the solution set of an inequality.

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution set. It shows all the possible values that satisfy the inequality.To graph our solution set, \([-\frac{1}{2}, 17.5]\), follow these steps:

• Identify the endpoints: -0.5 and 17.5.
• Place a closed circle (since the endpoints are included) at -0.5 and 17.5.
• Draw a line connecting these points, shading between the circles to indicate all the values in this range are part of the solution.

Graphing makes it clear which values are part of the solution set and enhances understanding of the inequality by providing a visual representation.

When combined with interval notation, graphing provides a complete picture of the solution, making it easier to comprehend and verify the result of the inequality.