Problem 8 - Algebra 1: Concepts and Skills

Question

Determine whether the inequality is a multi-step inequality. Then explain how you would solve the inequality. $$ 10-c \geq 6 $$

Step-by-Step Solution

Answer

The inequality \(10 - c \geq 6\) is a single-step inequality, not a multi-step one. The solution to the inequality is \(c \leq 4\).

Step 1: Identify the type of inequality

Firstly, recognize the type of inequality. By observation, it is evident that the equation \(10 - c \geq 6\) is a single-step inequality as 'c' appears only once, and it can be solved in one operation.

Step 2: Solve the inequality

To solve the inequality, isolate 'c' on one side. Implement subtraction, which reverses because of the nature of inequality. It will look like this: \(-c \geq 6 - 10\). Simplify it to \(-c \geq -4\).

Step 3: Final solution

Since we are dealing with a negative 'c', we need to change the direction of the inequality when multiplying or dividing by a negative number. So, we divide by -1 on both sides to isolate 'c', which gives: \(c \leq 4\).

Key Concepts

Algebraic Inequalities

When we talk about algebraic inequalities, we refer to mathematical statements in which two expressions are compared by an inequality sign: > (greater than), < (less than), \(\geq\) (greater than or equal to), or \(\leq\) (less than or equal to). Unlike equations, which show equality, inequalities show a range of possible solutions. For instance, in the inequality \(10 - c \geq 6\), it indicates that when 10 is reduced by 'c', the result is at least 6, which can also be read as 'c' is at most 4 after treating the inequality.

It is essential to remember that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality reverses. This is because the number line's order flips when considering negative values. Understanding this principle is crucial for correctly solving inequalities.

Isolating Variables

The process of isolating the variable is the key step in solving an inequality or equation. To 'isolate' means to get the variable on one side of the inequality sign all by itself. In the given problem \(10 - c \geq 6\), the variable 'c' is currently not by itself.

Subtraction to Isolate

To accomplish isolation, we often perform operations that 'undo' whatever is being done to the variable. In this case, we subtract 10 from both sides of the inequality, resulting in \( -c \geq -4\). The addition or subtraction principle of inequalities states you can add or subtract the same quantity from each side without affecting the inequality's direction.

Division to Isolate

Lastly, dividing both sides by -1 finalizes the isolation, giving us \(c \leq 4\). This step is accompanied by a reversal of the inequality sign, a rule that only applies when multiplying or dividing by negative numbers.

Inequality Solution Steps

The approach to solving an inequality is methodical and involves a series of inequality solution steps. Let's break down the essential steps:

• Identify: Determine the type of inequality and how many operations are necessary to isolate the variable. In our example, it's a single-step inequality.
• Solve: Carry out the required operation(s) to isolate the variable. We subtract 10 from both sides and then divide by -1, remembering to flip the inequality.
• Verify: Check the solution by substituting it back into the original inequality. Although not shown in the steps, it's always good practice to check your work.
• Express: Clearly state the answer. For \(10 - c \geq 6\), the solution is \(c \leq 4\), which reads 'c is less than or equal to 4'.

These steps ensure that the variable can be isolated correctly and the inequality can be solved accurately. Applying this structured approach helps students to tackle various algebraic inequalities with confidence.