Problem 30 - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line. $$ \frac{|2 n-7|}{3} \leq 0 $$

Step-by-Step Solution

Answer

The solution is \( n = \frac{7}{2} \).

Step 1: Understand the Properties of Absolute Value

The absolute value of a number is always greater than or equal to zero. Therefore, for the inequality \( \frac{|2n-7|}{3} \leq 0 \) to hold, \(|2n-7|\) must equal zero, because a positive divide by 3 cannot be less than zero.

Step 2: Set the Absolute Value Expression Equal to Zero

Since \(|2n-7| = 0\), we have that the expression inside the absolute value should also equal zero. Thus, we need to solve the equation:\[ 2n - 7 = 0 \]

Step 3: Solve for n

To find the value of \( n \), solve the equation from step 2:\[2n - 7 = 0 \2n = 7 = \frac{7}{2}\]

Step 4: Check the Solution

Substituting \( n = \frac{7}{2} \) back into the inequality:\[\\left| 2 \times \frac{7}{2} - 7 \right| = 0 \\frac{0}{3} \leq 0\] This verifies that \( n = \frac{7}{2} \) is indeed a solution.

Step 5: Graph the Solution Set on a Number Line

The solution to the inequality is the single point \( n = \frac{7}{2} \). On a number line, it is represented by a filled dot at \( \frac{7}{2} \), since that is the only value \( n \) can take.

Key Concepts

Absolute Value

When tackling inequalities involving absolute values, understanding the basic property of absolute value is key.

• The absolute value of a number is the distance between that number and zero on a number line. Hence, it is always non-negative, or \[ |x| \geq 0 \]
• In the original inequality \( \frac{|2n-7|}{3} \leq 0 \), the only way this inequality holds is when \(|2n-7| = 0\), because a positive number cannot be less than zero.

Thus, converting the inequality involved with absolute values into a simpler equation can aid in finding solutions efficiently. This means solving the equation \[ 2n - 7 = 0 \].By directly understanding these properties, you can streamline the solution process for absolute value inequalities.

Graphing on Number Line

Graphing the solution on a number line helps visualize the range of possible values for a variable. In the given example,

• The original inequality simplifies to an equation: \( 2n - 7 = 0 \).
• Solving for \( n \) gives the unique solution \( n = \frac{7}{2} \).
• This result means that there is only one point on the number line representing the solution, specifically \( n = \frac{7}{2} \).

To graph on a number line:- Place a filled dot at this point to indicate that \( n = \frac{7}{2} \) is included in the solution set.- Since no other values satisfy the condition, no other areas of the number line should be highlighted or shaded. Graphing inequalities with absolute values simplifies to point-specific plots when dealing with equivalent zero expressions.

Solution Verification

Verifying the solution to an inequality ensures that the derived solution actually satisfies the original equation.

• For \( n = \frac{7}{2} \), substitute back into \( |2n-7| \): \[ |2\times\frac{7}{2} - 7| = |7 - 7| = 0 \]
• Then confirm the original inequality holds: \[ \frac{0}{3} \leq 0 \], which simplifies to \[ 0 \leq 0 \]. This is true.

The solution verification process is a fundamental step that confirms accuracy. It involves retracing the steps with the obtained value to ensure no errors were made. Being thorough with verification helps in reinforcing the understanding and trust in the mathematical processes applied.