Problem 27 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(2 x-1>3\) and \(x+8 \leq 11\)

Step-by-Step Solution

Answer

The solution is \(2 < x \leq 3\), or in interval notation, \((2, 3].\)

Step 1: Solve the First Inequality

The first inequality is \(2x - 1 > 3\). Add 1 to both sides to isolate the term with \(x\): \[2x > 4\]. Next, divide both sides by 2 to solve for \(x\): \[x > 2\].

Step 2: Solve the Second Inequality

The second inequality is \(x + 8 \leq 11\). Subtract 8 from both sides to solve for \(x\): \[x \leq 3\].

Step 3: Combine the Solutions

The solutions must satisfy both inequalities. From Step 1, \(x > 2\) and from Step 2, \(x \leq 3\). Combining these, the solution is \(2 < x \leq 3\).

Step 4: Represent the Solution Graphically

Draw a number line. Shade the region between 2 and 3. Use an open circle at 2 to indicate that 2 is not included and a closed circle at 3 to show that 3 is included.

Step 5: Express the Solution in Interval Notation

The interval notation for the solution \(2 < x \leq 3\) is \((2, 3].\)

Key Concepts

Inequality Solving

When tackling compound inequalities like this one, the key is to address each inequality separately before combining results. For the given problem, we start with the first inequality, \(2x - 1 > 3\).
To isolate \(x\), you perform algebraic operations on both sides: add 1, then divide by 2.
This gives \(x > 2\).The second inequality is \(x + 8 \leq 11\).
By subtracting 8 from both sides, we determine \(x \leq 3\). A crucial part of solving compound inequalities is ensuring that both conditions are satisfied simultaneously.
For this problem, we need \(x\) values that fit both \(x > 2\) and \(x \leq 3\).Finally, we conclude that the solution is the intersection of these two sets, \(2 < x \leq 3\).
This highlights how solving inequalities often involves carefully performing operations to transform and then combine results.

Interval Notation

Interval notation offers a concise way to express the solution set of an inequality on a number line. For the solution \(2 < x \leq 3\), interval notation condenses this information into

• the left endpoint, denoted with an open bracket \((\), if not included,
• and a closed bracket \(]\) on the side of any included equality.

Thus, \((2, 3]\) succinctly captures that \(x\) is more than 2 but less than or equal to 3. Remember that with interval notation, a parenthesis \(()\) indicates that the boundary number is not part of the solution set, while a bracket \([]\) means it is included.
This system allows for quick communication of which values are part of the solution and which are not, without needing an extensive explanation.

Graphing Inequalities

Graphing inequalities visually represents solutions and helps clarify which parts of the number line are included in the solution set.To graph \(2 < x \leq 3\), start by drawing a standard number line.
You place key points on this line based on the inequality, marking both 2 and 3. For \(x > 2\), an open circle is drawn at 2 to show it's not part of the solution, while for \(x \leq 3\), a closed circle at 3 indicates inclusion.Shade the section of the number line between these two points to illustrate the entirety of possible solutions.

• This shaded area visually defines the solutions satisfying both parts of the compound inequality.

Through graphing, solutions become easier to understand and verify, reinforcing the logical approach used in algebraic manipulation.