Problem 66 - Elementary and Intermediate Algebra

Question

Solve each inequality. Write the solution set in interval notation and then graph it. $$ a+4-10 a>a-16 $$

Step-by-Step Solution

Answer

The solution is \(a < 2\) or \((-\infty, 2)\) in interval notation.

Step 1: Simplify both sides of the inequality

Start by simplifying both sides of the inequality. On the left side, combine like terms. Combine the terms with \(a\): \(a - 10a = -9a\). So the inequality becomes \(-9a + 4 > a - 16\).

Step 2: Move all terms involving \(a\) to one side

Next, we move all terms involving \(a\) to one side of the inequality. Subtract \(a\) from both sides: \(-9a + 4 - a > -16\). This simplifies to \(-10a + 4 > -16\).

Step 3: Isolate the term with the variable

Isolate \(-10a\) by subtracting 4 from both sides: \(-10a > -16 - 4\). This simplifies to \(-10a > -20\).

Step 4: Solve for \(a\)

To isolate \(a\), divide both sides by -10. Remember that dividing by a negative number reverses the inequality sign: \(a < \frac{-20}{-10}\), resulting in \(a < 2\).

Step 5: Write the solution set in interval notation

The solution \(a < 2\) in interval notation is \((-\infty, 2)\).

Step 6: Graph the solution on a number line

Draw a number line and shade the region to the left of 2. Use an open circle at 2 to indicate that 2 is not included in the solution set.

Key Concepts

Interval Notation

Interval notation is a method used to express the set of all solutions to an inequality. It uses intervals as a concise way to describe subsets of the real line. When we say "solution set," we mean the collection of numbers that make the inequality true.

For example, consider the inequality \(a < 2\). To express this using interval notation, we identify all the values of \(a\) that meet the condition.

• Since \(a\) can be any number less than 2 but not equal to 2, we use an open interval.
• Open intervals use parentheses to indicate that the endpoint is not included. In this case, the interval is \((-\infty, 2)\).
• The negative infinity symbol, \(-\infty\), is used because there is no lower bound to \(a\); it can be any number less than 2.

This notation tells us that the solution starts from \(-\infty\) and includes all numbers up to, but not including, 2.

Graphing Inequalities

Graphing inequalities involves visually representing the solution set on a number line. This helps us quickly understand which numbers are part of the solution.

To graph the inequality \(a < 2\), follow these steps:

• Draw a horizontal line, representing the number line.
• Mark the number 2 on the line. This helps us identify where to start thinking about the solution.
• Use an open circle at 2 to show that 2 is not included in the solution, as `less than` means it does not count the number itself.
• Shade the region to the left of the 2, indicating all numbers smaller than 2 satisfy the inequality.

By using this visual representation, you can easily see which numbers are less than 2. It acts as a quick check to confirm the interval notation \((-\infty, 2)\) matches the graphical solution.

Algebraic Simplification

Algebraic simplification is the process of rewriting expressions or equations to make them easier to work with. This is particularly useful when dealing with inequalities, as it helps isolate the variable in question.

Let's consider our original inequality: \(a + 4 - 10a > a - 16\). The goal is to simplify each side and put all terms involving \(a\) together. Here's how you do it step by step:

• Combine like terms: In this case, combine \(a\) and \(-10a\) on the left side to get \(-9a + 4\).
• Subtract \(a\) from both sides to gather \(a\) terms on one side: \(-10a + 4 > -16\).
• Isolate the variable term \(-10a\): Subtract 4 from both sides to get \(-10a > -20\).
• Divide by -10: To solve for \(a\), divide both sides by -10, and remember to reverse the direction of the inequality. This results in \(a < 2\).

These steps simplify the inequality, making it easier to determine the solution set both algebraically and graphically.