Problem 77 - College Algebra

Question

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$3|x-1|+2 \geq 8$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq -1\) or \(x \geq 3\). In interval notation, this would be expressed as \(-\infty, -1\] \cup [3, +\infty\)

Step 1: Isolate the Absolute Value Expression

Firstly, subtract 2 from both sides of the inequality to isolate the absolute value expression:\(3|x-1| \geq 6\).\nThen, divide both sides by 3 to make the coefficient 1: |x-1| \geq 2.

Step 2: Rewrite Without Absolute Value Bars

Express the absolute value inequality as two separate inequalities: \(x-1 \geq 2\) and \(x-1 \leq -2\). It's like thinking 'what distance can x be from 1 to be bigger than 2, either to the right or left of 1.'

Step 3: Solve Each Inequality

Now, solve each inequality separately:\nFor \(x-1 \geq 2\), add 1 to each side obtaining \(x \geq 3\)\nFor \(x-1 \leq -2\), add 1 to each side getting \(x \leq -1\)

Step 4: Represent the Solution on a Number Line and Interval Notation

Plot the numbers -1 and 3 on the number line. For the first inequality (\(x \leq -1\)), shade all numbers to the left of -1 (including -1). For the second inequality (\(x \geq 3\)), shade all numbers to the right of 3 (including 3). The interval notation for this is: \(-\infty, -1\] \cup [3, +\infty\)

Key Concepts

Inequality Solving

Solving inequalities, especially with absolute values, involves a systematic step-by-step approach. When you have an absolute value inequality like \( |x-1| \geq 2 \), you need to express it as two separate inequalities. Here’s why:

• Absolute Value Property: Absolute values denote distance from zero, which means \( |a| = b \) translates to \( a = b \) or \( a = -b \).
• Two Scenarios: For |x-1|, the distance from 1 must be greater than or equal to 2. This leads to the two situations, \( x - 1 \geq 2 \) and \( x - 1 \leq -2 \).

Next, you solve each inequality separately. Solve \( x - 1 \geq 2 \) by adding 1 to each side, leading to \( x \geq 3 \). This is the first part of the solution. For \( x - 1 \leq -2 \), also add 1 to each side, giving \( x \leq -1 \). This process helps convert the absolute value into comprehendible linear conditions to identify where the solution set falls.

Interval Notation

After solving the inequalities, it's important to express the solutions using interval notation. This notation is like a mathematical shorthand that quickly tells us where our solutions lie on a number line.Interval notation uses brackets:

• Closed Bracket \( \big[ \big] \): Includes the endpoint in the solution. Example: \([3, +\infty)\) includes 3.
• Open Bracket \( ( ) \): Excludes the endpoint from the solution. Example: \(-\infty, -1)\).

For our inequality, we found two segments: \( x \leq -1 \) and \( x \geq 3 \). In interval notation, list these as \((-\infty, -1] \cup [3, +\infty)\). The union symbol \(\cup\) means the solution is inclusive of both segments.

Number Line Graphing

Graphing inequalities on a number line offers a visual representation of solutions. With our problem, the solution \( x \leq -1 \lor x \geq 3 \) translates to specific areas on the line.To graph, follow these steps:

• Identify Points: Mark the points -1 and 3 on the line. These are crucial since they are boundaries for the solution segments.
• Shade Regions: For the segment \( x \leq -1 \), shade everything to the left of -1 and include -1 itself (use a closed circle).
• Separate Segments: There is a break or gap between \(-1\) and \(3\) (not shaded) which isn’t part of the solution. For \( x \geq 3 \), shade everything to the right of 3 including 3 (closed circle).

This visual graph aids quick understanding, coordinating with the interval notation: you can see that \((-\infty, -1] \cup [3, +\infty)\) maps perfectly with the shaded regions.