Problem 39 - Prealgebra

Question

Solve each inequality. Check your solution. $$2.9

Step-by-Step Solution

Answer

The solution to the inequality is \(c > -4.1\) or \((-4.1, \infty)\).

Step 1: Isolate the Variable

To solve the inequality \(2.9 < c + 7\), we need to isolate the variable \(c\) on one side. Start by subtracting 7 from both sides of the inequality: \(2.9 - 7 < c\). Simplifying the left side gives us \(-4.1 < c\). This means that \(c > -4.1\).

Step 2: Write Solution in Interval Notation

Since \(c > -4.1\), we can express the solution in interval notation as \(c \in (-4.1, \infty)\). This indicates that \(c\) can be any number greater than \(-4.1\).

Step 3: Verify the Solution

Let's check if our solution is correct by testing a value greater than \(-4.1\). If we use \(c = 0\), then substitute into the original inequality: \(2.9 < 0 + 7\). This simplifies to \(2.9 < 7\), which is true, confirming the solution.

Key Concepts

Solving Inequalities

To solve inequalities like \(2.9 < c + 7\), start by isolating the variable. Think about the inequality as a balance scale, and your goal is to keep the scale balanced while getting the variable by itself on one side. In this example, the inequality is \(2.9 < c + 7\). We need to move the 7 away from the \(c\) to have \(c\) alone.To achieve this, subtract 7 from both sides of the inequality. This looks like: \[ 2.9 - 7 < c + 7 - 7 \]Once simplified, it becomes:\[ -4.1 < c \]Now, the inequality is telling us that \(c\) must be greater than \(-4.1\). This process of solving inequalities is about maintaining equality on both sides while isolating the variable. It is similar to solving equations, but with an added focus on thinking about the direction of the inequality sign. Remember, if you ever multiply or divide an inequality by a negative number, you must flip the inequality sign.

Interval Notation

Once the inequality has been solved, it's often beneficial to express the solution using interval notation. This is a concise way to show the range of possible values a variable can take. Let's consider our solution \(c > -4.1\). To convey this inequality in interval notation:

• Identify the lower bound and consider that it is not included in the solution set (open interval), so you'll use a parenthesis.
• Since there is no upper limit to \(c\), it can go as far as infinity, which is also expressed as an open interval because infinity is not a reachable number.

Therefore, the interval notation is \((-4.1, \infty)\).This notation tells us exactly which values \(c\) can take, clearly showing that \(c\) is greater than \(-4.1\) and can increase indefinitely. Interval notation is a powerful tool because it eliminates ambiguity in expressing a range of values.

Checking Solutions

After solving an inequality, it's crucial to verify that your solution is correct. Checking your answer involves substituting a value from your solution set back into the original inequality. This helps ensure that the inequality holds true.In our case:

• The solution we found was \(c > -4.1\).
• Choose an easy-to-test number greater than \(-4.1\), such as 0.

Substitute \(c = 0\) into the original inequality: \[ 2.9 < 0 + 7 \]This simplifies to:\[ 2.9 < 7 \]This statement is true, confirming our solution. By checking solutions, we verify that our steps were correct and the values fit within the original conditions of the problem. Verifying is a great habit as it not only confirms your solution but also boosts your confidence in your algebra skills.