Problem 23 - Prealgebra

Question

Solve each inequality. Check your answer. $$-30 \leq d+(-5)$$

Step-by-Step Solution

Answer

The solution is \(-25 \leq d\) or \(d \in [-25, \infty)\).

Step 1: Isolate the Variable

Start with the inequality \(-30 \leq d + (-5)\). The first step is to isolate the variable \(d\) on one side of the inequality. To do this, add \(5\) to both sides of the inequality to eliminate the \(-5\) on the right side. This gives:\[-30 + 5 \leq d\]Simplify the left side:\[-25 \leq d\].

Step 2: Write the Solution

The inequality \(-25 \leq d\) tells us that the solution is all values of \(d\) that are greater than or equal to \(-25\). This can be written as the interval \([-25, \infty)\).

Step 3: Check the Solution

To verify the solution, choose a value for \(d\) from the solution set. If \(d = -25\), substitute it back into the original inequality:\[-30 \leq -25 + (-5)\]Simplify the right side:\[-30 \leq -30\]This is a true statement, confirming that \(d = -25\) satisfies the inequality. To further verify, choose another value, for example \(d = 0\), and substitute it:\[-30 \leq 0 + (-5)\]which simplifies to:\[-30 \leq -5\]This is also true, confirming that the solution set \([-25, \infty)\) is correct.

Key Concepts

Solving Inequalities

Solving inequalities involves finding the range of values that satisfy a given inequality statement. Unlike equations, which depict exact equality, inequalities describe a relationship where one side could be less than, greater than, or equal to the other side. For our problem, we start with the inequality \[-30 \leq d + (-5)\], which means that the value on the left is less than or equal to the expression on the right.

Our goal in solving inequalities is to isolate the variable (\(d\) in this case) on one side to reveal its possible values. We achieve this by performing operations that simplify the inequality without changing its meaning. These operations should maintain the inequality's order, such as adding or subtracting the same quantity from both sides.

In the solution, we added 5 to both sides to eliminate \(-5\): \[-30 + 5 \leq d\], simplifying further to \[-25 \leq d\]. This tells us that \(d\) must be greater than or equal to \(-25\). Expressed in interval notation, this translates to \([-25, \infty)\), where all values from \(-25\) to infinity satisfy the inequality.

Mathematical Reasoning

Mathematical reasoning is the logical process of thinking through a math problem to devise a coherent strategy and solution. When working with inequalities, understanding the properties of inequality symbols and arithmetic operations on inequalities is crucial.

Consider the direction of the inequality: in our problem, the inequality \(-30 \leq d + (-5)\) means "less than or equal to." We must consistently apply operations that maintain this direction to solve the inequality correctly.

When adding or subtracting numbers, the inequality's direction remains unchanged. However, be aware that multiplying or dividing by a negative number reverses the inequality sign. This principle doesn't apply here, but it's essential for solving other inequalities.

By adding 5 to both sides, we used mathematical reasoning to simplify and isolate the variable while keeping the inequality valid and maintaining the logical order. Such conceptual understanding ensures precise problem-solving and enables verifying whether the solution aligns with the original inequality.

Checking Solutions

After solving an inequality, it is important to verify that the solution is correct. Checking solutions to inequalities ensures that no mistakes were made during the calculation process.

To check the solution of \(-25 \leq d\), we substitute specific values from the solution set back into the original inequality \(-30 \leq d + (-5)\). Choose convenient values, like the boundary value (\(-25\)) or any arbitrary value that easily computes, such as \(0\).

• When \(d = -25\), substituting gives \[-30 \leq -25 + (-5)\], simplifying to \[-30 \leq -30\], which holds true.
• When \(d = 0\), substituting results in \[-30 \leq 0 + (-5)\], simplifying to \[-30 \leq -5\], which is also valid.

These checks confirm that any \(d\) greater than or equal to \(-25\) satisfies the inequality, validating the solution set \([-25, \infty)\). By consistently checking solutions, any possible errors are caught, and confidence in the solution's accuracy is established.