Problem 13 - Prealgebra

Question

Solve each inequality. Check your answer. $$t+6>-3$$

Step-by-Step Solution

Answer

The solution is \( t > -9 \).

Step 1: Isolate the variable on one side

To solve the inequality \( t + 6 > -3 \), we need to isolate the variable \( t \). Start by subtracting 6 from both sides of the inequality to get rid of the constant term on the left-hand side. This gives us \( t + 6 - 6 > -3 - 6 \).

Step 2: Simplify both sides

Simplify the expressions on both sides of the inequality: on the left, \( t + 6 - 6 \) simplifies to \( t \), and on the right, \( -3 - 6 \) simplifies to \( -9 \). So, the simplified inequality is \( t > -9 \).

Step 3: Check the solution

To check our solution, pick a value for \( t \) that satisfies the inequality \( t > -9 \). Let's choose \( t = -8 \). Substitute \( t = -8 \) back into the original inequality: \( -8 + 6 > -3 \). This simplifies to \( -2 > -3 \), which is true. This confirms the solution is correct.

Key Concepts

Isolating Variables

When dealing with inequalities, isolating the variable is a crucial step. It helps you view clearly what values the variable can take. In the inequality \( t + 6 > -3 \), we need to get \( t \) on its own on one side of the inequality sign.

• Identify the Term to Remove: Notice the constant \(+6\) added to \(t\). This is what needs to be removed to isolate \(t\).
• Perform the Opposite Operation: To remove \(+6\), subtract 6 from both sides of the inequality.
• Write the New Inequality: After subtracting, our inequality becomes \(t > -9\).

This process keeps the inequality balanced and correctly represents the solutions for \(t\). Isolating variables is similar to solving equations, but always remember: if you multiply or divide by a negative number, you must reverse the inequality sign.

Checking Solutions

After solving an inequality, it is important to check if the solution meets the original inequality condition. This helps ensure that no errors were made during the solving process. For the inequality \( t > -9 \), here's how you check:

• Select a Test Value: Choose any value for \( t \) that is greater than \(-9\). For example, \( t = -8\).
• Substitute and Simplify: Replace \( t \) with \(-8\) in the original inequality: \(-8 + 6 > -3\).
• Verify the Inequality: Calculate \(-2 > -3\), which is a true statement.

The truth of this statement confirms that our solution is correct. Checking solutions is essential because it reassures you that the operations performed during solving were valid and accurate.

Simplifying Expressions

Simplifying expressions is fundamental to solving both equations and inequalities. It helps reduce complex problems into simpler ones that are easier to work with. In our solved inequality, once you isolate the variable by subtracting 6, you need to simplify both sides:

• Left Side Simplification: \( t + 6 - 6\) simplifies to \( t\) because \(+6\) and \(-6\) cancel each other out.
• Right Side Simplification: Perform the arithmetic \(-3 - 6\) to obtain \(-9\).

Simplifying expressions is about eliminating unnecessary terms and combining terms wherever possible, making the inequality straightforward and manageable. This process not only clarifies what the inequality looks like when solved, but also prevents errors and confusion.