Question

Find the solution set of the disjunction \(2-3 x>5\) or \(2 x-1>5\)

Short answer

The solution set for the disjunction \(2-3x > 5\) or \(2x-1 > 5\) is \(x < -1\) or \(x > 3\).

Step-by-step solution

Solve the first inequality \(2-3x > 5\)

To solve the inequality, we want to isolate x. Follow these steps: 1. Subtract 2 from both sides: \(-3x > 3\) 2. Divide both sides by -3 (remember to flip the inequality sign when dividing by a negative number): \(x < -1\)

Solve the second inequality \(2x-1 > 5\)

To solve the inequality, we want to isolate x. Follow these steps: 1. Add 1 to both sides: \(2x > 6\) 2. Divide both sides by 2: \(x > 3\)

Find the union of the solution sets

We now have the two inequalities: \(x < -1\) and \(x > 3\). Since we are looking for the disjunction (the "or" statement), we will take the union of their solution sets. The solution set for the disjunction is \(x < -1\) or \(x > 3\). This means that any value of x less than -1 or greater than 3 will satisfy at least one of the inequalities.

Key concepts

Disjunction

In mathematics, a disjunction is a logical operation that combines two statements with the word "or." When solving inequalities, a disjunction helps us find the range of values that satisfy either of the inequalities.

For example, in the exercise above, we deal with the disjunction of two inequalities:

• The first inequality is: \( 2 - 3x > 5 \)
• The second inequality is: \( 2x - 1 > 5 \)

This means we are looking for values of \( x \) that make either one or both of these inequalities true.

So when you see a conjunction with "or," it allows for more flexibility in solutions because values can satisfy either condition. It's not necessary for \( x \) to meet both conditions, just one is enough.

Solution Set

The term solution set refers to all possible values that satisfy a given condition or inequality. When solving inequalities, we determine the solution set by isolating the variable and finding out what values make the inequality true.

Let's discuss how to find the solution sets for the individual inequalities given in the exercise:

• For the first inequality, \( 2 - 3x > 5 \), we find that the solution set is \( x < -1 \).
• For the second inequality, \( 2x - 1 > 5 \), the solution set is \( x > 3 \).

Each part of a disjunction has its own solution set. Once you have them, you can explore further by evaluating the union, which gives the combined set of satisfying values for the entire inequality.

Union of Solution Sets

When working with a disjunction in inequalities, like "or," we need to consider the union of solution sets. This means combining the solution sets of each inequality to find all possible solutions.

The union of solution sets incorporates any value of \( x \) that makes at least one of the inequalities true. In our example:

• The first solution set is \( x < -1 \).
• The second solution set is \( x > 3 \).

When you take their union, the solution set becomes \( x < -1 \) or \( x > 3 \). This means any number less than -1 or greater than 3 will satisfy the original disjunction.

The union allows us to fully express all solutions, accommodating both inequalities, and is a crucial step in solving disjunctions.