Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Copy and complete the statement using the correct inequality symbol. If \(-\frac{3}{4} x \geq-12\), then \(x\)_________16.

Step-by-Step Solution

Answer

If \(-\frac{3}{4} x \geq-12\), then \(x \leq 16\).

Step 1: Initial Inequality

Start with the given inequality: \(-\frac{3}{4} x \geq-12\). We need to solve for \(x\).

Step 2: Solve for x

Divide both sides of the inequality by -3/4 to find the value of \(x\). Remember that dividing or multiplying by a negative number when dealing with inequalities changes the direction of the inequality sign. Therefore, -12 divided by -3/4 is equal to 16 and the sign becomes less than or equal to (≤).

Step 3: Final Answer

After isolating \(x\) on one side of the inequality we get: \(x \leq 16\) as our final answer.

Key Concepts

Inequality Symbols

When comparing quantities in mathematics, we use inequality symbols to show the relationship between them. The most common symbols are: less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). It's crucial to choose the correct symbol to accurately represent the relationship. For example, if one quantity is larger than another, we'd use '>'. If they could be equal or if one is larger, we'd use '≥'. Understanding these symbols is essential for expressing and solving algebraic inequalities effectively.

Improving clarity on this topic could involve creating visual aids that depict various quantities on a number line, showing the appropriate inequality symbol for the relationship between those quantities. Additionally, real-world examples help students draw connections between the abstract symbols and concrete quantities they encounter in daily life.

Algebraic Inequalities

Algebraic inequalities, such as \( -\frac{3}{4} x \geq-12 \) present in the exercise, involve finding the set of all possible values of a variable that make the inequality true. Solving these inequalities typically follows the same steps as solving equations: isolating the variable on one side. However, a critical difference is how the inequality sign changes when both sides of the inequality are multiplied or divided by a negative number. These steps require careful attention to detail to ensure the final inequality correctly reflects the relationship between the quantities involved.

For deeper understanding, learners might benefit from practice problems with stepwise feedback. This offers them the chance to correct misconceptions and reinforces the procedural knowledge needed to manage the sign changes and other nuances of solving inequalities.

Dividing by a Negative Number

A key concept in understanding algebraic inequalities is knowing what happens when we divide by a negative number. In our exercise, dividing both sides of \( -\frac{3}{4} x \geq-12 \) by the negative number \( -\frac{3}{4} \) results in the inequality sign changing from 'greater than or equal to' to 'less than or equal to'. This rule is counter-intuitive and a common source of errors for students. The reason for this 'flip' of the inequality sign is tied to the number line; negative numbers are to the left of zero, so when we divide by a negative number, we essentially 'flip' the direction of the inequality.

Providing students with interactive exercises where they can manipulate the inequality and see the effects of dividing by negative numbers in real-time may significantly enhance their comprehension. Using visual tools, such as number lines, to show this 'flip' can also make the concept more tangible.