Problem 29 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$\frac{m}{2}+9 \geq 5$$

Step-by-Step Solution

Answer

The solution is \(m \geq -8\). Graphically, a closed circle at -8 with an arrow extending to the right.

Step 1: Subtract 9 from Both Sides

To isolate the term with the variable \(m\), subtract 9 from both sides of the inequality. This yields: \(\frac{m}{2} + 9 - 9 \geq 5 - 9\), which simplifies to \(\frac{m}{2} \geq -4\).

Step 2: Multiply Both Sides by 2

To solve for \(m\), multiply both sides of the inequality by 2 to eliminate the fraction. This gives \(2 \times \frac{m}{2} \geq 2 \times (-4)\). Simplifying both sides produces \(m \geq -8\).

Step 3: Check the Solution

To verify, choose a value for \(m\) that is greater than or equal to -8, such as \(-8\), and substitute back into the original inequality: \(\frac{-8}{2} + 9 \geq 5\). Simplifying, we get \(-4 + 9 \geq 5\), which is true since 5 is equal to 5.

Step 4: Graph the Solution on a Number Line

Represent the solution \(m \geq -8\) on a number line. Use a closed circle at -8 to indicate that -8 is included in the solution set, and draw a line extending to the right to represent all numbers greater than -8.

Key Concepts

Solving Inequalities

When solving inequalities, the process is very similar to solving regular equations. First, you want to isolate the variable on one side. Let’s break down the procedure using our given inequality:

Start by subtracting constants from both sides to get the term with the variable alone. For example, with \( \frac{m}{2} + 9 \geq 5 \), subtract 9 from each side to obtain \( \frac{m}{2} \geq -4 \).
Next, if the variable is tied up in a fraction, clear it by multiplying both sides by the denominator. Here, multiply each side by 2 to solve for \( m \), thus \( m \geq -8 \).

• Step 1: Isolate the variable with subtraction or addition.
• Step 2: Eliminate fractions via multiplication or division.

A crucial point to remember: when you multiply or divide by a negative number, you must flip the inequality symbol. However, this wasn't necessary in our example.

Graphing Inequalities

Graphing inequalities gives a visual representation of the solution set. Here's how to do it:

To graph the inequality \( m \geq -8 \), you start by marking -8 on a number line. For inequalities that include the boundary value (like \( \geq \) or \( \leq \)), use a closed circle to signify that this endpoint is included in the solution.
When the inequality symbol is less than or greater than, without equality (\( < \) or \( > \)), use an open circle instead.
Draw an arrow starting from the circle. For \( m \geq -8 \), draw it to the right, covering all numbers greater than -8. Here's a simple rule:

• Use a closed circle for \( \leq \) and \( \geq \).
• Use an open circle for \( < \) and \( > \).
• Point the arrow in the direction of greater (right) or less (left) based on the inequality.

This makes sure anyone can quickly identify the solution set on the line.

Checking Solutions

Checking your solution after solving an inequality ensures its correctness. It involves substituting a number from the solution set back into the original inequality.

For the inequality \( \frac{m}{2} + 9 \geq 5 \), our solution was \( m \geq -8 \). To check, pick \( m = -8 \) as it’s part of the solution set. Substitute it back:
to become \( \frac{-8}{2} + 9 \geq 5 \). Simplifying further: \( -4 + 9 \geq 5 \), which equates to \( 5 \geq 5 \) and holds true.Remember these steps for checking solutions:

• Choose a value from the solution set.
• Substitute this value back into the original inequality.
• Verify the inequality holds true with substitution.

If the substituted value satisfies the inequality, your solution is correct. Thus, checking solutions like this builds confidence in your results.