Problem 46 - Elementary and Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ 4.5 x-1<-10 \text { or } 6-2 x \geq 12 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -2)\) and is represented by an open interval on a number line.

Step 1: Solve the First Inequality

The first inequality is \(4.5x - 1 < -10\). To solve it, add 1 to both sides to get \(4.5x < -9\). Then, divide both sides by 4.5 to solve for \(x\): \(x < -2\).

Step 2: Solve the Second Inequality

The second inequality is \(6 - 2x \geq 12\). Subtract 6 from both sides to get \(-2x \geq 6\). Divide both sides by -2 and reverse the inequality sign, yielding \(x \leq -3\).

Step 3: Combine the Solutions

We have \(x < -2\) from the first inequality and \(x \leq -3\) from the second inequality. Since this is an 'or' compound inequality, we take the union of the solution sets.

Step 4: Express the Solution in Interval Notation and Graph

For \(x < -2\), the interval is \((-\infty, -2)\). For \(x \leq -3\), the interval is \((-\infty, -3]\). The union of these intervals, \((-\infty, -2)\) and \((-\infty, -3]\), is simply \((-\infty, -2)\). On a number line, this is expressed as an open circle at -2, indicating that -2 is not included, and shading to the left.

Key Concepts

Solving Inequalities

Solving inequalities involves finding the range of values that satisfy a given inequality. In the provided problem, two inequalities need to be solved: one with a less than sign and the other with a greater than or equal to sign.

To solve each inequality, we follow these basic steps:

• Isolate the variable, usually by performing arithmetic operations on both sides.
• For example, in the inequality \(4.5x - 1 < -10\), we begin by adding 1 to both sides, simplifying to \(4.5x < -9\). To find \(x\), we then divide both sides by 4.5, resulting in \(x < -2\).
• In the case of \(6 - 2x \geq 12\), we subtract 6 from both sides to obtain \(-2x \geq 6\). Because we're dividing by a negative number (-2), we reverse the inequality, yielding \(x \leq -3\).
• It’s essential to remember that when you multiply or divide an inequality by a negative number, you need to flip the inequality sign!

Combining solutions in compound inequalities depends on the logical operator ("or" or "and"), resulting in different solution sets.

Interval Notation

Interval notation is a concise way to state a set of numbers, especially for representing the solution of inequalities.

This notation uses brackets and parentheses to show closed and open intervals:

• A parenthesis \((\) or \()\) indicates that the endpoint is not included. For example, \((-fty, -2)\) represents all numbers less than -2, without including -2 itself.
• A bracket \([\) or \()]\) indicates inclusion of the endpoint. In our example, \((-fty, -3]\) means all numbers less than or equal to -3, including -3.

In compound inequalities with an "or" statement, we find the union of the intervals obtained from each inequality. For example, the union of \((-fty, -2)\) and \((-fty, -3]\) results in \((-fty, -2)\). It represents numbers satisfying either of the inequalities.

Graphing Solution Sets

Graphing solution sets helps visualize the range of values that satisfy an inequality. It involves marking intervals on a number line according to the solution.

Here's how you can create such a graph:

• Identify the critical points and mark them on the number line. For \(x < -2\), mark an open circle at -2, denoting that -2 is not included.
• Similarly, for \(x \leq -3\), mark a closed circle at -3, indicating inclusion of -3.
• Shade the region to the left of -2 to represent all numbers less than -2.
• In this case, the union of \(x < -2\) and \(x \leq -3\) is depicted by the shading extending infinitely leftward from -2, excluding -2 itself.

The graph visually clarifies the abstract solution, making it easier to understand compound inequalities.