Problem 27 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$-10 x<40$$

Step-by-Step Solution

Answer

The solution to the inequality -10x < 40 is \(x > -4\). This is graphed with an open circle at -4 and an arrow to the right.

Step 1: Isolate x

The first step is to isolate x from the inequality. The coefficient of x in the inequality is -10 and to isolate x, the inequality needs to be divided by -10. Remember: when you divide or multiply an inequality by a negative number, the inequality sign changes direction. Hence, the inequality by dividing by -10 becomes \(x > -4 \).

Step 2: Graph the inequality

After solving the inequality, the next step is to graph the inequality on a number line. At x=-4, draw an open circle because x is not equal to -4. Draw an arrow to the right from the open circle to show x is greater than - 4.

Key Concepts

Graphing Inequalities

Graphing inequalities helps us visually understand the range of possible solutions. When you graph an inequality like \( x > -4 \), it's important to illustrate it on a number line accurately. Here's how you do it:

• Identify the critical point from the solved inequality, which is \( x = -4 \) in this case.
• Place an open circle on \( x = -4 \). An open circle means that the value itself is not included in the solution.
• Draw an arrow pointing to the right starting from \(-4\) to indicate that the solutions are all numbers greater than \(-4\).

This visual representation helps you see where all the solutions of the inequality lie, allowing for a clearer understanding of the problem.

Inequality Sign Changes

Understanding why and when inequality signs change is critical when solving inequalities. In this exercise, dividing both sides of the inequality \(-10x < 40\) by a negative number \(-10\) changes the inequality sign. Here's why:

• When you multiply or divide both sides of an inequality by a positive number, the inequality direction remains unchanged.
• However, multiplying or dividing by a negative number reverses the inequality. So, \(-10x < 40\) becomes \(x > -4\) after dividing by \(-10\).

The reversal of the inequality sign is key to correctly solving and graphing the inequality. Failing to reverse the sign will lead to incorrect solutions.

Number Line Representation

Representing inequalities on a number line provides an easy-to-understand visual. It shows all possible solutions, making it easier to comprehend how inequalities work.

• Start by placing an open or closed circle at the value extracted from the inequality (\(-4\) in this case). Use an open circle because \(x\) does not equal \(-4\).
• Identify the direction of the inequality. For \(x > -4\), draw an arrow to the right, showing all greater values are included.
• This visual representation simplifies complex mathematical concepts, aiding better understanding for visual learners.

Using a number line turns abstract mathematical expressions into something concrete and relatable, enhancing comprehension and retention.