Problem 9 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing a single operation on both sides. State the operation, indicating whether or not the inequality changes direction. Solve the inequality. $$ -4.1+c \leq 2.3 $$

Step-by-Step Solution

Answer

Answer: The range of values for "c" is c ≤ 6.4, meaning that "c" is less than or equal to 6.4.

Step 1: Identify the operation needed to solve the inequality

In this exercise, we need to solve the inequality, -4.1 + c ≤ 2.3. In order to isolate the variable "c" on one side of the inequality, we will have to perform the operation of addition. Specifically, we will add 4.1 to both sides, which will not change the direction of the inequality.

Step 2: Add 4.1 to both sides of the inequality

Now, let's add 4.1 to both sides of the inequality: $$ -4.1 + c + 4.1 \leq 2.3 + 4.1 $$

Step 3: Simplify the inequality

The inequality simplifies to: $$ c \leq 6.4 $$ Now, we have found the range of values for "c": c is less than or equal to 6.4.

Key Concepts

Solving Inequalities

When tackling inequalities, the goal is often to get the variable by itself on one side of the inequality. This process involves using operations that maintain the equation's balance, similar to solving equations. However, inequalities have their special rules you must always consider. One important thing to remember is that while you are reorganizing the inequality by applying operations, the type of operation and the numbers involved may influence the solution's direction or even require a change in the inequality sign. Always keep these rules in mind, and you'll smoothly manage the inequalities.

Direction of Inequality

Understanding the direction of inequality is key in working with inequalities. The inequality might use symbols like \(<\), \(>\), \(\leq\), or \(\geq\), which tell you the relationship between the sides.

• The direction indicates whether the expression on the left is less than or greater than the expression on the right.
• When solving, most operations won't change the direction unless you multiply or divide by a negative number. This is a vital aspect that separates handling inequalities from equations.
• In our case, adding positive numbers doesn't affect the direction, so \(-4.1 + c \leq 2.3\) stays in the same direction after manipulating it to \(c \leq 6.4\).

Recognizing this fixed direction helps ensure your solution maintains the right logical relationship between the expressions.

Addition Operation in Inequalities

The addition operation is one of the simplest ways to solve inequalities, particularly when your target is to isolate a variable. When you add the same number to both sides, it's similar to keeping a balanced scale even. Here's how it works:

• Consider \( -4.1 + c \leq 2.3 \). You want \(c\) by itself, so you decide to add 4.1, which cancels the negative number beside \(c\).
• This gives both sides the expression \( -4.1 + 4.1 + c \leq 2.3 + 4.1 \).
• The left side simplifies to \( c \), and the right side becomes \( 6.4 \).

Adding a constant helps to isolate the variable without flipping the inequality's direction as long as the constant isn't negative. Always double-check the operation to confirm it was applied correctly and verify the final inequality stays accurate.