Problem 49 - Prealgebra

Question

Solve each inequality. Check your solution. $$-9+k>20$$

Step-by-Step Solution

Answer

The solution is \( k > 29 \).

Step 1: Isolate the variable

To isolate the variable \( k \), we need to apply the inverse operation to move \( -9 \) to the other side of the inequality. This means we will add 9 to both sides of the inequality. So, the inequality becomes \( k > 20 + 9 \).

Step 2: Simplify the expression

Now, simplify the right side of the inequality by performing the addition: \( 20 + 9 = 29 \). Therefore, the inequality becomes \( k > 29 \).

Step 3: Check the solution with a test value

Choose a value for \( k \) that satisfies \( k > 29 \). Let's take \( k = 30 \) as a test value. Substitute \( k = 30 \) into the original inequality: \(-9 + 30 > 20\), which simplifies to \( 21 > 20 \). This is a true statement, confirming that our solution is correct.

Key Concepts

Understanding Inequalities

When we talk about inequalities in prealgebra, we're looking at comparisons between two expressions using symbols to show how they relate to each other. The most common inequality symbols are:

• ">": greater than
• "<": less than
• "≥": greater than or equal to
• "≤": less than or equal to

Inequalities are similar to equations, but instead of finding where two expressions are equal, we're finding where one expression is larger or smaller. So, while an equation gives you one specific answer, an inequality gives you a range of possible answers where the inequality holds true.
When working with inequalities, it's important to remember that reversing the direction of the inequality sign occurs when multiplying or dividing both sides by a negative number. This rule is key to maintaining the true relationship between the expressions.

Steps in Solving Inequalities

Solving inequalities often involves steps similar to solving equations, but with some additional rules. Here’s a basic walkthrough:

• Isolate the variable: Just like with equations, the first step is to get the variable by itself on one side of the inequality. In our exercise, to isolate the variable "k", we needed to add 9 to both sides.
• Simplify the inequality: Next, perform any simplifications to find a clearer solution. In our example, adding 9 to 20 gives us 29, so we have "k > 29".
• Test a solution: Finally, choose a test value to ensure your solution is correct. Here, by trying "k = 30" and substituting it back into the original inequality, we saw that it holds true as "21 > 20".

Remember, testing your solution helps verify that none of the steps involved incorrect transformations, especially when dealing with signs.

Algebra Concepts in Inequalities

Inequalities are a fundamental part of algebra. They involve basic algebraic skills like moving terms, performing arithmetic operations, and analyzing relationships between numbers. Here's how these algebra concepts come into play:

• Moving terms: This involves adding or subtracting terms from both sides of an inequality to isolate the variable, similar to doing these actions in equations.
• Performing arithmetic operations: Involves adding, subtracting, multiplying, or dividing both sides of an inequality by the same number, keeping track of how these operations affect the inequality sign, especially if the number is negative.
• Understanding relationships: Recognizing how changes to the variable affect the feelings of greater than, less than, or equal can help in visualizing the correct range of solutions.

By practicing these algebra concepts, you build a deeper understanding of not only how to solve inequalities but also how to apply these fundamental principles to more complex math problems.