Problem 15 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$9+2 p \leq 15$$

Step-by-Step Solution

Answer

The solution is \( p \leq 3 \). On a number line, it is shown by a filled circle at 3 and shading to the left.

Step 1: Isolate the variable term

Start by subtracting 9 from both sides of the inequality: \[ 9 + 2p - 9 \leq 15 - 9 \]. This simplifies to: \[ 2p \leq 6 \].

Step 2: Solve for the variable

Divide both sides of the inequality by 2 to isolate \( p \): \[ \frac{2p}{2} \leq \frac{6}{2} \]. This simplifies to: \[ p \leq 3 \].

Step 3: Check the solution

To check the solution, choose a value for \( p \) that satisfies \( p \leq 3 \), for example, \( p = 3 \). Substitute \( p = 3 \) back into the original inequality: \[ 9 + 2 \times 3 = 9 + 6 = 15 \]. Since \( 15 \leq 15 \), the solution holds true.

Step 4: Graph the solution

To graph \( p \leq 3 \) on a number line, draw a number line with points labeled around 3. Draw a solid circle at 3 to indicate that 3 is included in the solution set and shade everything to the left of 3 to represent all numbers less than or equal to 3.

Key Concepts

Solving Inequalities

Inequalities are mathematical expressions that show the relationship between two values. Solving an inequality is similar to solving an equation, with an extra step: maintaining the inequality sign. When you're asked to solve an inequality like \( 9 + 2p \leq 15 \), you want to find the range of values for the variable \( p \) that makes the inequality true.

• **Step 1**: **Isolate the variable**: Move terms around so all terms with the variable are on one side. For \( 9 + 2p \leq 15 \), you subtract 9 from both sides to obtain \( 2p \leq 6 \).
• **Step 2**: **Solve for the variable**: Divide every term by the coefficient of the variable. So, dividing by 2 gives \( p \leq 3 \).
• **Important**: If you multiply or divide by a negative number, remember to flip the direction of the inequality.

Remember, solving inequalities provides a range of solutions, not just an exact number. It's crucial to check a number from that range, ensuring it satisfies the initial inequality.

Graphing Inequalities

Visualizing inequalities on a number line offers insight into all possible solutions. Suppose you've solved an inequality and found \( p \leq 3 \). You would represent this on a number line to reflect all values that meet the inequality criteria.

• Draw a horizontal line with numbers labeled appropriately around the solution.
• Identify the critical point, in this instance, place a solid circle on 3 since \( p \) can equal 3.
• Shade the portion of the line to the left of 3 to indicate all numbers less than or equal to 3. This shading illustrates the entire set of numbers making the inequality true.

Drawing on a number line confirms your solutions and underscores the inequality's range.

Prealgebra Concepts

Prealgebra is the foundational skill set for managing numbers through mathematical operations. It prepares you for algebra by instilling the competency to handle equations and inequalities. Let's explore the essential components that come into play when dealing with inequalities.

• **Understanding operations**: Prealgebra teaches addition, subtraction, multiplication, and division, needed to manipulate equations as seen in solving \( 9 + 2p \leq 15 \).
• **Variables and expressions**: A variable (like \( p \)) represents unknown values which you solve by isolating and calculating.
• **Logical thinking**: Recognizing the implications of an inequality assists in making predictions and checking solutions.

Building these prealgebra skills not only readies students for algebra but also boosts confidence in solving real-world problems with mathematical logic.