Problem 41 - Algebra 1

Question

SOLVING INEQUALITIES Solve the inequality. $$11.2 x \leq 134.4$$

Step-by-Step Solution

Answer

The solution of the inequality is \(x \leq 12\). Therefore, any value of \(x\) that is less than or equal to 12 will satisfy the inequality.

Step 1: Start with the given inequality

Write down the initial inequality which is \(11.2x \leq 134.4\). This equation states that 11.2 times some number \(x\) is less than or equal to 134.4.

Step 2: Divide both sides by 11.2

Divide each side of the equation by 11.2 to solve for \(x\). This simplifies the left side to just \(x\) because \(11.2/11.2\) is 1, and \(1*x = x\). The right side becomes \(134.4/11.2\).

Step 3: Compute the division

Perform the division on the right side. \(134.4\) divided by \(11.2\) equals to \(12\). The inequality is now solved and can be written as \(x \leq 12\).

Key Concepts

Linear Inequalities

Linear inequalities like the one in our problem, \(11.2x \leq 134.4\), are expressions where one side is not simply equal to the other but instead is less than, less than or equal to, greater than, or greater than or equal to.

Just like linear equations, these inequalities feature terms of the first degree such as \(11.2x\). However, instead of an equality sign (\(=\)), they use inequality symbols, for instance \(\leq\), \(\geq\), \(<\), or \(>\).

Solving them involves finding all possible values of \(x\) that make the inequality true. When these values are plotted on a number line, it shows us a range of solutions, unlike a single point as observed in equations.

Inequality Solution Steps

When tackling an inequality, it is useful to follow a structured approach:

• **Start with the given inequality**: We're dealing with \(11.2x \leq 134.4\). Identify the parts — coefficient \(11.2\), variable \(x\), and constant \(134.4\).
• **Simplify or manipulate**: The goal is to isolate \(x\). To do that, any multiply or divide operations, such as dividing by \(11.2\), help to simplify.
• **Calculation**: Calculate accurately. For instance, dividing \(134.4\) by \(11.2\) gives \(12\).
• **Conclusion**: Present the final solution. Here, it's \(x \leq 12\).

Step-by-step methods ensure that you accurately solve the inequality and understand the steps leading to the solution.

Inequality Division Method

One of the key techniques in solving inequalities is the division method. This helps isolate the variable term. Consider our example, \(11.2x \leq 134.4\).

We divide each side by \(11.2\) to remove the coefficient from \(x\). This leads to: \[ x \leq \frac{134.4}{11.2} \]

Crucially, when dividing or multiplying inequalities by a positive number, the inequality sign remains unchanged. If it was a negative number, we would reverse the inequality sign. Here, with \(11.2\) being positive, the sign remains \(\leq\), leading to \(x \leq 12\). Simplifying the division successfully gives us the solution.

Simplification in Inequalities

Simplification is the process of reducing an inequality to its simplest form, making it easier to understand and solve. In this problem, simplification involved handling the multiplication aspect: \(11.2x\).

Dividing both sides by \(11.2\) simplifies the expression by canceling out terms efficiently. After performing the division, \(x\) is neatly isolated with its coefficient gone, turning the problem into \(x \leq 12\).

During simplification, it's important to ensure calculations are precise to maintain the inequality's integrity and arrive at the correct solution. This step is essential for understanding and solving any inequality thoroughly.