Problem 36 - Elementary and 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

Interval notation: (3, 9)

Step 1: Solve the Left Inequality

Start by solving the left part of the double inequality: \( 7 < 3x - 2 \). Add 2 to both sides to isolate the term with \( x \). This gives \( 9 < 3x \). Then, divide each side by 3 to solve for \( x \): \( 3 < x \).

Step 2: Solve the Right Inequality

Next, solve the right part of the double inequality: \( 3x - 2 < 25 \). Add 2 to both sides to isolate the term with \( x \). This gives \( 3x < 27 \). Then, divide each side by 3 to solve for \( x \): \( x < 9 \).

Step 3: Combine Inequalities and Write Interval Notation

Combine the results from Step 1 and Step 2 to write the solution as a single compound inequality: \( 3 < x < 9 \). In interval notation, this is written as \((3, 9)\).

Step 4: Graph the Solution Set

To graph the solution set \((3, 9)\), draw a number line. Place an open circle on 3 and an open circle on 9. Shade the region between these two points to represent all numbers greater than 3 and less than 9, excluding 3 and 9 themselves.

Key Concepts

Graphing Inequalities

Graphing inequalities is a crucial skill in understanding how a range of values can satisfy an inequality condition. The process involves visualizing the set of possible solutions on a number line. Consider the double inequality from our exercise: \(3 < x < 9\). When graphing this:

• Draw a horizontal line representing the number line.
• Identify the critical values in the inequality - in this case, 3 and 9.
• Place open circles at these points. Open circles indicate that these end-values are not included in the solution set.
• Shade the region between the two circles to show all numbers greater than 3 and less than 9 are included.

Open circles are used because the inequality is strict (\(<\), not \(\leq\)). If it were non-strict, you would use closed circles. By graphing, you visually understand which values make the inequality true.

Compound Inequalities

A compound inequality consists of two separate inequalities joined by either "and" or "or". The original exercise demonstrates a type of compound inequality known as a double inequality, where a single variable is sandwiched between two numbers, such as \(7 < 3x - 2 < 25\).

• "And" compound inequalities specify that both conditions must be true simultaneously. The solution set consists of values that satisfy both inequalities. In this case, that's between 3 and 9.
• "Or" compound inequalities would allow for solutions to satisfy either condition, leading to possibly disjoint sets of solutions.

Understanding compound inequalities is key in determining the overlap or connection between different sets of possible solutions, leading to a comprehensive solution range.

Interval Notation

Interval notation is a concise way to express the set of all real numbers that satisfy an inequality. For our exercise, the solution to the double inequality \(3 < x < 9\) is expressed in interval notation as \((3, 9)\). Here's how interval notation works:

• Parentheses, \(( )\), indicate that an endpoint is not included in the interval. This is the case when the inequality is strict.
• Brackets, \([ ]\), indicate that an endpoint is included. You would use brackets if the inequality were non-strict, like \(\leq\).
• The lower bound value is written first, followed by the upper bound, separated by a comma.

Using \((3, 9)\) tells us that the solution consists of all numbers between 3 and 9, not including those boundary values. Interval notation provides clarity and simplicity in mathematics by compactly summarizing solution sets.