Problem 52 - Algebra 2

Question

CHALLENGE Graph each set on a number line. a. \(-2 < x < 4\) b. \(x < -1\) or \(x > 3\) c. \((-2 < x < 4)\) and \((x < -1 \text { or } x >3)\) (Hint: This is the intersection of the graphs in part a and part b. d. Solve \(3 < |x+2| \leq 8 .\) Explain your reasoning and graph the solution set.

Step-by-Step Solution

Answer

Graph the solutions on a number line for each part, considering open and closed points where specified.

Step 1: Analyzing Part (a)

We need to graph the inequality \(-2 < x < 4\) on a number line. This means that \(x\) must take on values between -2 and 4, not including -2 and 4. On the number line, this includes an open circle at -2 and an open circle at 4, with a shaded line in between to indicate all the numbers in that interval.

Step 2: Analyzing Part (b)

For \(x < -1\) or \(x > 3\), we graph two separate sections on the number line. The first section includes all numbers less than -1, with an open circle at -1 extending to the left. The second section includes all numbers greater than 3, with an open circle at 3 extending to the right.

Step 3: Analyzing Part (c)

This part involves finding the intersection of the graphs from parts (a) and (b). Intersection means the numbers where both conditions from part (a) and part (b) are satisfied. Looking at the inequalities \(-2 < x < 4\) and \(x < -1\) or \(x > 3\), the values that satisfy both are those where \(x < -1\) (since they don't overlap within the primary range), other sections within \(x > 3\) to 4 apply.

Step 4: Solving Part (d)

This challenge requires solving the compound inequality \(3 < |x+2| \leq 8\). This absolute value inequality splits into two separate inequalities: \(x+2 > 3\) and \(x+2 < -3\) for the first part, and \(x+2 \leq 8\) and \(x+2 \geq -8\) for the second part. Solving these, we have two sets: 1. **For** \(x+2 > 3\): \(x > 1\). 2. **For** \(x+2 < -3\): \(x < -5\). 3. **For** \(x+2 \leq 8\): \(x \leq 6\). 4. **For** \(x+2 \geq -8\): \(x \geq -10\). Combined, the solution is \(x < -5\) or \(1 < x \leq 6\). Graphically, you plot an open circle at -5 extending left and a line segment from 1 (open circle) to 6 (closed circle).

Key Concepts

Number Line

A number line is a simple yet powerful tool that helps illustrate numbers and the relationships between them visually. It's a straight horizontal line with each point representing a real number.

• Positive numbers are placed to the right of zero.
• Negative numbers are located to the left of zero.
• An open circle on the line indicates that a number is not part of the solution set, while a closed circle shows it is included.

Number lines are especially helpful when graphing inequalities because they show the range of possible solutions conveniently. For example, when graphing o numbers from does not include and demonstrating

Intersection of Inequalities

The intersection of inequalities refers to the common solution set that satisfies multiple conditions at the same time. When dealing with inequalities like or we need to identify the range of values where both conditions overlap. These intersections can be visualized easily on a number line.

• Look for overlaps in shaded regions or line segments.
• The intersection includes only the values that meet all given inequalities.

For instance, when graphing the intersection of and from parts (a) and (b) in the original exercise, you find certain intervals where both conditions apply, resulting in a more restrictive range of values.

Absolute Value Inequalities

Absolute value inequalities involve expressions under an absolute value sign, like or These represent the distance of a number from zero on the number line and can have two potential outcomes since distance is always positive.

• To solve, split the inequality into two separate cases.
• Account for both the positive and negative possibilities.

For example, with into two separate inequalities: and and solve them individually to find the complete solution set. Always check your solutions on a number line to ensure they reflect all possible scenarios.

Compound Inequalities

Compound inequalities are combinations of two or more inequalities, like and These can be joined using "and" or "or," which defines how their solution sets relate to each other.

• "And" involves finding a solution that satisfies all inequalities.

Compound Inequalities

Compound inequalities are combinations of two or more inequalities, like and These can be joined using "and" or "or," which defines how their solution sets relate to each other.

• "And" involves finding a solution that satisfies all inequalities.
• "Or" provides a solution set that satisfies at least one of the inequalities.

For the exercise at hand, consider the inequality which combines previous solutions to set a condition where either one part or the other must be true. This approach is helpful when analyzing complex restrictions and visualizing them graphically can clarify how the conditions interact on a number line.