Problem 55 - Elementary and Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ -x<-2 x \text { and } 3 x>2 x $$

Step-by-Step Solution

Answer

The solution set is empty, expressed as \(\emptyset\).

Step 1: Solve the first inequality

The first inequality is \(-x < -2x\). To simplify, add \(-2x\) to both sides to get \(-x + 2x < 0\). This simplifies to \(x < 0\).

Step 2: Solve the second inequality

The second inequality is \(3x > 2x\). Subtract \(2x\) from both sides to get \(x > 0\).

Step 3: Determine the intersection

We have \(x < 0\) and \(x > 0\) from the two inequalities. There is no number that satisfies both conditions simultaneously.

Step 4: Graph the solution set and write in interval notation

Since there is no number that satisfies both \(x < 0\) and \(x > 0\) simultaneously, the solution set is empty. On a number line, this would be represented by no shading area. The interval notation for this is \(\emptyset\).

Key Concepts

inequality solving

Inequality solving involves finding the set of values for a variable that makes an inequality true. Inequalities are expressions that use symbols like <, >, ≤, or ≥. Solving an inequality is very similar to solving an equation. The goal is to isolate the variable on one side to determine its possible values. When solving compound inequalities, which involve two inequalities usually combined by "and" or "or," you need to tackle each inequality separately before determining their combined solution. For example, when faced with

• -x < -2x
• 3x > 2x

You would solve each one on its own. The key is to perform operations that simplify the inequality. Remember to reverse the inequality sign whenever you multiply or divide by a negative number. This is crucial to obtaining the correct values for the variable.

interval notation

Interval notation is a way to represent a set of numbers, often solutions of inequalities, in a compact form. It uses brackets and parentheses to denote intervals on the real number line.

• Square brackets [ ] are used when the endpoint is included in the solution, known as a closed interval.
• Parentheses ( ) are used when the endpoint is not included, indicating an open interval.

For compound inequalities like those in the problem

• If the inequalities intersect, you write the overlapping part as the interval.
• If there is no overlap, as with solutions like empty set, it is represented as \( \emptyset \).

Interval notation is a clean and efficient way of conveying the range of values that satisfy an inequality. It helps avoid lengthy explanations and ensures clarity in mathematical communication.

graphing inequalities

Graphing inequalities involves representing the solution to an inequality on a number line. Each solution is depicted visually, which can help in understanding the range of possible answers.To graph x < 0 and x > 0:

• Mark a circle at the number but do not fill it in; this indicates the number is not included (open circle).
• Shade the arrow extending to the values that satisfy the inequality.

For compound inequalities, particularly those with "and" conjunctions, you're interested in the intersection of their solutions on the number line. When both conditions, such as \(x < 0\) and \(x > 0\) in the given problem, have no numbers satisfying both, the graph has no shaded area. This illustrates that the solution is an empty set, \( \emptyset \), confirming visually that no values can simultaneously satisfy both conditions.