Problem 14 - Prealgebra

Question

Solve each inequality. Check your solution. $$15 \geq 3 t$$

Step-by-Step Solution

Answer

The solution is \(t \leq 5\).

Step 1: Isolate the Variable

To solve the inequality \(15 \geq 3t\), begin by isolating the variable \(t\). Divide both sides of the inequality by 3 to achieve this:\[\frac{15}{3} \geq \frac{3t}{3}\]This simplifies to:\[5 \geq t\]

Step 2: Interpret the Solution

The inequality \(5 \geq t\) tells us that \(t\) can be any number that is less than or equal to 5.

Step 3: Check the Solution

To verify the solution, choose a value for \(t\) that satisfies \(5 \geq t\). For example, try \(t = 4\):Substitute \(t = 4\) back into the original inequality:\[15 \geq 3(4)\]Calculate:\[15 \geq 12\]Since 15 is indeed greater than or equal to 12, \(t = 4\) is a valid solution. You can try other values such as \(t = 5\) or \(t = 0\) to ensure they also satisfy the inequality.

Key Concepts

Understanding Prealgebra

Prealgebra is the foundational math field that prepares students for algebra by building essential skills. It's like the training wheels before learning to ride a bike. In prealgebra, you encounter basic arithmetic and introduction to variables, equations, and inequalities.

With prealgebra, you begin to see how numbers and operations come together to form expressions. Problems usually involve simple mathematical operations such as addition, subtraction, multiplication, and division. You'll also learn how to perform these operations on variables, not just numbers. Understanding these operations is crucial as they lay the groundwork for algebraic thinking. Here’s why it’s important:

• Prealgebra helps in developing reasoning and logical thinking.
• It forms the basis for higher mathematics.
• Improves problem-solving abilities applicable to real-life situations.

As you can see, a solid grasp of prealgebra sets you up nicely to tackle more complex algebraic concepts with confidence.

Approaching Math Problem Solving

Math problem solving is a critical skill that goes beyond just being computational. It's about understanding problems, analyzing them, and coming up with a strategy to solve them. In the context of solving inequalities like the one in our problem, it follows a structured approach.

• **Read and understand the problem.** Understand what is given and what you need to find.
• **Set up a plan.** Determine which mathematical operations or methods will help you solve the problem.
• **Execute the plan.** Perform calculations carefully while following mathematical rules.
• **Review the solution.** Check if your answer makes sense in the context of the problem.

When you solve an inequality, you're not just finding a solution but identifying a range of possible solutions. This requires careful analysis and verification to ensure all possible solutions fit the original condition. Remember, practice and patience are your best friends in honing your problem-solving skills.

Variable Isolation Techniques

Variable isolation involves rearranging an equation or inequality to get the variable alone on one side. This process makes it easier to find the value of the variable. It's a fundamental technique in algebra and is essential for solving equations and inequalities.

• **Identify the variable you need to isolate.** Look at the inequality and find the variable.
• **Perform inverse operations.** Use operations that reverse the equations' actions. For instance, if the variable is multiplied, divide to isolate it.
• **Keep the balance.** Whatever you do on one side of the equation, you must do on the other to maintain equality or inequality.

In the original exercise, dividing both sides of the inequality by 3, we isolated the variable \(t\). This gives us an easier inequality \(5 \geq t\) to interpret. Understanding and applying these steps are crucial in tackling more complex algebraic expressions.

Mastering Inequality Solutions

Inequality solutions can seem tricky, but they are a logical extension of solving equations. The ultimate goal is to find all values of the variable that make the inequality true. Here’s how you can approach them effectively:

• **Solve the inequality as you would an equation.** Start with isolating the variable.
• **Graph the solution if needed.** This can help visualize the set of possible solutions.
• **Check your solution.** Substitute values back into the original inequality to verify correctness.
• **Be mindful of inequality direction.** Multiplying or dividing both sides by a negative number flips the inequality sign.

The problem \(15 \geq 3t\) simplifies to \(5 \geq t\), indicating \(t\) can be any number less than or equal to 5. This simple example illustrates how once you understand the concept, you can apply it to solve and check more complicated inequalities effectively.