Problem 51 - Prealgebra

Question

Solve each inequality. Check your solution. $$12+z \leq 8$$

Step-by-Step Solution

Answer

The solution to the inequality is \( z \leq -4 \).

Step 1: Identify the Inequality

The inequality presented is \(12 + z \leq 8\). Our task is to solve for \(z\).

Step 2: Isolate the Variable

To solve for \(z\), first subtract \(12\) from both sides of the inequality: \[ 12 + z - 12 \leq 8 - 12 \] This simplifies to \[ z \leq -4 \].

Step 3: Verification Step

Verify the solution by substituting \(z = -4\) back into the original inequality: \[ 12 + (-4) \leq 8 \] Simplifying this gives \[ 8 \leq 8 \], which is true. Thus, the solution \(z \leq -4\) is correct.

Key Concepts

Inequality solutions

When solving inequalities, such as the one provided in this exercise, our primary goal is to find all values of a variable that make the inequality true. Inequalities are similar to equations, but they indicate a range of possible solutions rather than a single answer. In this particular problem, we are dealing with the inequality \(12 + z \leq 8\), where the inequality sign "\(\leq\)" indicates that the expression on the left should be less than or equal to the expression on the right.
To solve the inequality, we can use techniques very similar to solving equations. Here, we're interested in isolating the variable \(z\). By performing operations that keep the inequality balance, like subtracting \(12\) from both sides, we found that \(z \leq -4\).
Thus, the solution to the inequality is any value for \(z\) that is \(-4\) or smaller. This means that there is not one specific answer but a set of answers, represented on a number line as all points from \(-4\) extending to negative infinity.

Verification process

Verifying the solution to an inequality involves substituting the solution back into the original inequality to check for correctness. This ensures we haven't made any mistakes during the arithmetic operations. In this exercise, our solution was \(z \leq -4\).
To verify, we substitute \(z = -4\) back into the original inequality \(12 + z \leq 8\). After substitution, we get \(12 + (-4) \leq 8\), which simplifies to \(8 \leq 8\). This statement holds true, confirming that our solution is correct.
The verification process provides confidence that the steps followed were accurate and that the solution covers the required range. In real-world problems, this step is essential as it ensures the integrity of results especially when the inequality forms part of a larger problem.

Math problem solving

Math problem solving involves a systematic approach to finding solutions. When dealing with inequalities, the process includes understanding the problem, finding a method to solve it, executing the method, and then checking the results. Each phase is critical to achieving the correct solution.
Understanding the inequality and its components is the first step. Identifying both sides of the inequality helps in determining the steps needed to isolate the variable. In our exercise, recognizing which operations to use allowed us to simplify \(12 + z \leq 8\) into \(z \leq -4\).
Once a solution is found, verification is key. By checking that our answer satisfies the inequality, we avoided potential mistakes from earlier steps. Additionally, understanding the context of a math problem, such as recognizing whether it is seeking a range of values rather than a single solution, allows for deeper comprehension and application in different scenarios. Thus, effective problem solving involves not only finding the solution but also understanding and applying the right techniques.