Problem 36 - Intermediate Algebra

Question

Solve each double inequality. Graph the solution set and write it using interval notation. \(7<3 x-2<25\)

Step-by-Step Solution

Answer

The solution in interval notation is \((3, 9)\).

Step 1: Solve the First Inequality

The given double inequality is \(7 < 3x - 2 < 25\). Start by solving the first inequality \(7 < 3x - 2\). Add 2 to both sides to get \(9 < 3x\). Next, divide both sides by 3 to isolate \(x\), resulting in \(3 < x\).

Step 2: Solve the Second Inequality

Now solve the second inequality \(3x - 2 < 25\). Again, add 2 to both sides to obtain \(3x < 27\). Divide both sides by 3, giving \(x < 9\).

Step 3: Combine the Solutions

Combine the solutions from the first and second inequalities. The solutions of \(3 < x\) and \(x < 9\) can be combined to \(3 < x < 9\). This is the solution to the double inequality.

Step 4: Graph the Solution Set

Graph the solution \(3 < x < 9\) on a number line. Open circles should be placed at 3 and 9 to indicate that these endpoints are not included in the solution set, and draw a line between them to indicate that all numbers in between are solutions.

Step 5: Write the Solution in Interval Notation

Express the solution \(3 < x < 9\) in interval notation. Since 3 and 9 are not included, use parentheses: the solution is written as \((3, 9)\).

Key Concepts

Inequalities

Inequalities are mathematical expressions or equations where two values or expressions are compared using inequality signs such as \(<\), \(>\), \(\leq\), and \(\geq\). In this context, a double inequality involves two inequalities combined together, such as \(a < x < b\), which states that \(x\) is greater than \(a\) and less than \(b\).

In the given exercise, the double inequality \(7 < 3x - 2 < 25\) needs to be solved to find the range of values that \(x\) can take. We solve this by handling each part of the inequality separately. Solving double inequalities involves applying similar arithmetic operations on both sides of the inequality, like addition, subtraction, multiplication, or division—carefully keeping track of the direction of the inequality sign. It's important to remember that multiplying or dividing both sides of an inequality by a negative number will reverse the direction of the inequality sign.

Interval Notation

Interval notation is a shorthand way of writing a set of numbers, typically which form a solution set of an inequality. This notation uses parentheses \(()\) and brackets \([]\) to describe intervals on a number line.

When you have a result from a solved inequality such as \(3 < x < 9\), interval notation makes it easy to express these bounds without the need of words.

• Parentheses \(( \) or \()\) indicate that the endpoint is not included.
• Brackets \([ \) or \()\) show that the endpoint is included.

In the given exercise, the solution \(3 < x < 9\) is expressed in interval notation as \((3, 9)\). The use of parentheses here tells us that neither 3 nor 9 are included in the solution set.

Graphing Inequalities

Graphing inequalities visually represents a range of solutions on a number line or in a coordinate plane. In the case of a double inequality like \(3 < x < 9\), you would mark the number line to show which parts are part of the solution set.

To graph the solution on a number line:

• Place open circles at the endpoints, 3 and 9, to show these numbers are not included in the solution set.
• Draw a line between these open circles to represent all numbers between 3 and 9 are within the solution set.

This graphical representation helps to quickly understand which values satisfy the inequality and makes it easier to see the proportion of the number line affected by the solution.