Problem 55 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

No solution exists; the solution set is \(\emptyset\).

Step 1: Solve the first inequality

Consider the first inequality \(-x < -2x\). We can solve it by adding \(2x\) to both sides to get \(x < 0\).

Step 2: Solve the second inequality

Next, let's solve the second inequality \(3x > 2x\). Subtract \(2x\) from both sides to yield \(x > 0\).

Step 3: Analyze the solution

Now we have two inequalities: \(x < 0\) and \(x > 0\). Interpret these to see if a common solution exists. The inequality \(x < 0\) means all numbers less than zero, while \(x > 0\) means all numbers greater than zero. There is no number that is both less than and greater than zero at the same time.

Step 4: Graph and interval notation

On a number line, \(x < 0\) and \(x > 0\) do not overlap; hence, there is no solution that satisfies both inequalities. In interval notation, the solution is an empty set, represented as \(\emptyset\).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line by using intervals. It provides a shorthand way to express which numbers are included in a set.

When you solve inequalities, interval notation helps you communicate the solution set effectively.

• For example, the interval \((-\infty, 0)\) represents all numbers less than 0. It includes every number from negative infinity up to, but not including, 0. Notice that the parenthesis "(" means 0 is not part of the solution.
• Similarly, the interval \((0, \infty)\) includes all positive numbers greater than 0, again excluding 0.

To describe a set with no solution, like when inequalities have no numbers in common, we use the empty set notation, \(\emptyset\). This means there are no numbers that satisfy both inequalities. Interval notation helps succinctly convey this information.

Solution Set

The solution set is a collection of numbers that satisfies all parts of an inequality or system of equations.

When dealing with compound inequalities, you are looking for numbers that fulfill more than one condition at the same time.

• Take, for example, the inequalities \(x < 0\) and \(x > 0\). The task here is to find numbers that can solve both inequalities.
• However, in our example, there are no numbers that are both less than and greater than zero simultaneously.

Thus, the solution set for these compound inequalities is empty, expressed as \(\emptyset\). It signifies that there are no valid solutions that meet all the conditions in the inequalities.

Inequality Graphing

Inequality graphing is a visual representation of solutions to inequalities on a number line. It helps in quickly identifying the range of numbers that satisfy a given inequality.

In graphing inequalities:

• Open dots are used to represent numbers that are excluded from the solution set.
• Shaded lines or arrows indicate the range of numbers included.

For the inequalities \(x < 0\) and \(x > 0\), you would draw a line representing numbers less than 0 to the left of 0, using an open dot on 0. Similarly, draw another line representing numbers greater than 0, to the right, also with an open dot on 0.

Since these lines do not meet or overlap anywhere on the number line, they visually indicate there is no common solution, reinforcing the idea of an empty set \(\emptyset\) for the solution.