Problem 46 - Prealgebra

Question

Solve each inequality. Check your solution. $$-5 m \geq-15$$

Step-by-Step Solution

Answer

The solution is \( m \leq 3 \).

Step 1: Isolate the Variable Term

Start by moving the coefficient of the variable to the other side. We need to get rid of -5 from the left-hand side of the inequality. Divide both sides of the inequality by -5, remembering to reverse the inequality sign whenever dividing by a negative number: \[ m \leq \frac{-15}{-5} \] Simplifying the right side gives us: \[ m \leq 3 \]

Step 2: Verify the Solution

To verify the solution, pick a number less than or equal to 3 and substitute it back into the original inequality to see if it holds true. Let's check with \( m = 3 \): \[ -5(3) \geq -15 \] Calculating, we get: \[ -15 \geq -15 \] This is true. Thus, \( m \leq 3 \) is a correct solution.

Key Concepts

Understanding Prealgebra and its Application

Prealgebra forms the foundational language of mathematics that most people first engage with. It covers basic arithmetic and introduces concepts like variables, expressions, and equations. One of the key skills in prealgebra is solving equations and inequalities, which are mathematical statements showing the relationship between quantities that may be equal or unequal.

In prealgebra, an inequality differs from an equation primarily in that the values are not equal, but one is greater or smaller than the other. For example, solving an inequality such as \(-5m \geq -15\) involves isolating the variable, just like in equations, but with an important twist that relates to the direction of inequality signs.

As part of prealgebra, students must grasp how to manipulate inequalities using rules that apply when integers, variables, or fractions are involved. These rules help bring forth mathematical reasoning and facilitate more advanced learning.

Building Mathematical Reasoning Skills

Mathematical reasoning is the logical thought process required to solve math problems efficiently. It involves understanding how to use mathematical concepts and formulas to get to the correct solution.

In our inequality \(-5m \geq -15\), we start by isolating the variable. By doing this, we're applying mathematical reasoning to simplify the inequality and find the possible values for \(m\). Here's how reasoning comes into play:

• We divide both sides by \(-5\) to cancel out the multiplicative factor attached to \(m\).
• Whenever we divide or multiply both sides by a negative number, we remember to reverse the inequality sign to maintain the truth of the inequality.

These steps highlight how reasoning helps us understand not just how to perform operations but why they'd lead us to the right answer.

Developing reasoning skills is essential for all mathematical topics as it improves problem-solving strategies, enhances our ability to think critically, and lays a solid grounding for algebraic concepts.

Basics of Inequality Verification

Verification in mathematics means checking that a solution is correct. When we solve inequalities, verifying the result is crucial to ensure our solution is valid in the given inequality context.

Let's revisit our example: once we isolate the variable to find \(m \leq 3\), the verification step involves substituting numbers back into the original inequality to see if they hold true. By choosing \(m = 3\) and substituting back into \(-5m \geq -15\), we confirm the answer because \(-15 \geq -15\) is true. This check tells us our solution is correct.

• Begin by substituting the boundary value. If it holds true, try other possible numbers within the solution set.
• Verification helps catch errors like forgetting to reverse the inequality sign when dividing by a negative number.

These checks ensure the integrity of the solution process, giving confidence that results align with mathematical principles.