Problem 41 - Intermediate Algebra

Question

Solve each compound inequality. Graph the solution set and write it using interval notation. \(3 x+2<8\) or \(2 x-3>11\)

Step-by-Step Solution

Answer

The solution is \((-\infty, 2) \cup (7, \infty)\).

Step 1: Solve the Inequality 1: \(3x + 2 < 8\)

The first inequality to solve is \(3x + 2 < 8\). Start by isolating \(x\). Subtract 2 from both sides to get \(3x < 6\). Next, divide each side by 3 to solve for \(x\): \(x < 2\).

Step 2: Solve the Inequality 2: \(2x - 3 > 11\)

The second inequality provided is \(2x - 3 > 11\). Start by adding 3 to both sides: \(2x > 14\). Then, divide each side by 2 to solve for \(x\): \(x > 7\).

Step 3: Graph the Solution Set

To graph the solution set for \(x < 2\) or \(x > 7\), draw a number line. For \(x < 2\), use an open circle on 2 and shade to the left. For \(x > 7\), use an open circle on 7 and shade to the right. The sections shaded represent the solution set.

Step 4: Write the Solution in Interval Notation

Since the solution is a union of two inequalities, the interval notation that describes this compound inequality is \((-\infty, 2) \cup (7, \infty)\).

Key Concepts

Solving Inequalities

When it comes to solving inequalities, it's important to think of them as puzzles where you isolate the variable to find possible values that satisfy the inequality. In the given exercise, there are two separate inequalities to solve:

• First: \(3x + 2 < 8\)
• Second: \(2x - 3 > 11\)

To tackle the first inequality, subtract 2 from both sides to get \(3x < 6\) and then divide by 3, resulting in \(x < 2\).For the second inequality, add 3 to both sides which gives \(2x > 14\). Then divide by 2, finding that \(x > 7\).The method of adjusting the inequality while maintaining balance is similar to solving equations. Remember, if you ever multiply or divide both sides by a negative number, the direction of the inequality sign will flip. Thankfully, in the given exercise, this isn't necessary.

Interval Notation

Interval notation is a shorthand way to describe ranges of possible values for a variable. This is very handy when dealing with solutions that include a range of numbers, like in compound inequalities. For our inequalities \(x < 2\) and \(x > 7\), we use interval notation to express these as:

• \((-infty, 2)\) for x < 2. This notation signifies all numbers less than 2 but not including 2 itself.
• \((7, infty)\) for \(x > 7\). This covers all numbers greater than 7 but not including 7 itself.
• To combine these into one expression that accounts for either condition being true (since it's connected by "or"), we use the union symbol \(\cup\): \((-infty, 2) \cup (7, \u007finfty)\).

This notation elegantly communicates that solutions to the compound inequality fall in either of these non-overlapping intervals.

Graphing Solutions

Graphing the solutions of inequalities allows a visual representation of all possible solutions on a number line. To represent \(x < 2\) and \(x > 7\), follow these steps:

• First, draw a simple horizontal number line.
• Next, identify the numbers 2 and 7 on the line.
• For the inequality x < 2, place an open circle over 2. This signifies that 2 itself is not part of the solution set.
• Shade to the left starting from this open circle, covering all numbers leading to negative infinity.
• Similarly, for x > 7, place another open circle on 7. Like before, this means 7 is not included in the solutions.
• Shade to the right from this open circle to indicate all numbers greater than 7 lead towards positive infinity.

Thus, two separate shaded areas on the number line emerge. This graph visually underscores the solution expressed in interval notation.