Problem 28 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 1<3 x+4 \leq 16 $$

Step-by-Step Solution

Answer

The solution is \(( -1, 4 ]\) and should be graphed between \(-1\) and \(4\), excluding \(-1\).

Step 1: Understand the Inequality Structure

The inequality presented is a double inequality: \[1 < 3x + 4 \leq 16\]This means we have two separate inequalities to solve that are combined into one: 1. \(1 < 3x + 4\) 2. \(3x + 4 \leq 16\)

Step 2: Solve the First Inequality

To solve the inequality \(1 < 3x + 4\), subtract 4 from both sides:\[1 - 4 < 3x\] Which simplifies to:\[-3 < 3x\] Divide both sides by 3 to solve for \(x\):\[-1 < x\] or equivalently \(x > -1\).

Step 3: Solve the Second Inequality

To solve the inequality \(3x + 4 \leq 16\), subtract 4 from both sides:\[3x + 4 - 4 \leq 16 - 4\]Which simplifies to:\[3x \leq 12\]Divide both sides by 3:\[x \leq 4\].

Step 4: Combine the Solutions

The solution to the compound inequality is the intersection of the two results from the previous steps:\(x > -1\) and \(x \leq 4\).Thus, the combined solution is: \(-1 < x \leq 4\).

Step 5: Express Solution in Interval Notation

The solution \(-1 < x \leq 4\) can be written in interval notation as:\(( -1, 4 ]\)

Step 6: Graph the Solution Set

To graph the solution set on a number line:- Draw a number line.- Place an open circle at \(-1\) because the interval does not include \(-1\).- Place a closed dot at \(4\) because the interval includes \(4\).- Shade the region between \(-1\) and \(4\) to indicate all numbers in this range are solutions.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It simplifies expressing the solution set of inequalities and makes them visually interpretable. In our exercise, the solution to the inequality was

• \(-1 < x \leq 4\) which states that \(x\) is greater than \(-1\) but less than or equal to 4.

The interval notation for this solution is \((-1, 4]\). This means:

• The parenthesis "(" next to \(-1\) indicates \(-1\) is not included in the solution set.
• The square bracket "]" by the 4 signifies that 4 is part of the solution set.

Interval notation helps in translating the inequalities' range into a concise format. This is especially useful when dealing with multiple solutions in a compound setting.

Compound Inequality

A compound inequality consists of two or more inequalities joined by "and" or "or". In simple terms, it's like combining two mathematical statements. Our task was to solve the compound inequality \( 1 < 3x + 4 \leq 16\).

• "And" indicates that both conditions must be true at the same time, as seen in this case.

The solution process involves handling each inequality separately and then finding the overlap, known as the intersection, of their solutions. For instance:

• From \(1 < 3x + 4\), we got \(x > -1\), meaning \(x\) must be greater than \(-1\).
• From \(3x + 4 \leq 16\), we found \(x \leq 4\), indicating \(x\) must be less than or equal to 4.

So, the concept of a compound inequality is crucial in interpreting the overlapping section, \(-1 < x \leq 4\), where both these conditions are satisfied.

Graphing Inequalities

Graphing inequalities visually represents solutions on a number line, making it easier to comprehend and communicate solutions. To graph the inequality \(-1 < x \leq 4\):

• First, draw a horizontal line, which will serve as your number line.

• Add a small open circle at \(-1\) to show that \(-1\) is not included.
• Place a closed dot at 4, indicating that 4 is part of the solution.

Finally, shade in the region between \(-1\) and 4 to display all numbers within this range are part of the solution set. Using a graph allows quick visual confirmation of the solution, helping you see where the inequality extends on the number line. This abstract way to interpret the inequality solutions aids in recognizing valid \(x\) values.