Question

equation (2) \(x=-3\) (d) \((x \geq-1) \cup(x=-3)\)

Short answer

Answer: The solution set includes all values of \(x\) that are greater than or equal to \(-1\) and \(x\) equal to \(-3\).

Step-by-step solution

Solve Equation (2)

Equation (2) is straightforward: \(x=-3\). This means the solution is just \(x=-3\).

Understand Equation (d)

Equation (d) is a compound inequality involving a union: \((x \geq-1) \cup(x=-3)\). A union means that the solution set will include all values that satisfy either of the inequalities.

Solve the individual inequalities

First, solve the inequality \(x \geq -1\). It's already solved, so the solution is just \(x \geq -1\). Next, solve \(x = -3\). The solution is just \(x = -3\).

Combine solutions

Since we have a union, the solution set includes all values that satisfy either inequality. So, the solution for (d) is \((x \geq-1) \cup(x=-3)\) which includes all values of \(x\) that are greater than or equal to \(-1\) and \(x\) equal to \(-3\).

Key concepts

Compound Inequalities

Compound inequalities are mathematical expressions that combine two or more inequalities. To solve them, we often use the word "and" or "or". For instance,

• If we say "and," we must satisfy both conditions simultaneously.
• Using "or" means satisfying at least one of the conditions.

This makes "or" key to understanding the union of inequalities. In the expression \((x \geq -1) \cup (x = -3)\), the word "or" is implied by the union symbol \(\cup\). This tells us to find values of \(x\) that meet any one of these criteria. We solve each inequality separately and then combine the results, considering only the parts that meet at least one condition.

Solution Set

A solution set is the collection of all possible values that satisfy a given equation or inequality. When dealing with compound inequalities, identifying the solution set is crucial. Let's take the statement \((x \geq -1) \cup (x = -3)\) as an example.

• First, address each part: solving \(x \geq -1\) gives us all \(x\) values that are -1 or greater.
• The other part, \(x = -3\), means \(x\) is strictly -3.

The solution set must consider both solutions. Since this is a union, it includes the values from each individual solution. Therefore, the entire set will typically be a range or list of numbers that fit the conditions outlined.

Union of Inequalities

The union of inequalities is a way to combine solutions from different inequalities into one set. Represented by the symbol \(\cup\), it includes all solutions from each inequality. So in a compound inequality like \((x \geq -1) \cup (x = -3)\), we look for numbers that solve either, or both, inequalities.

• This includes any number greater than or equal to -1.
• Additionally, it includes the number -3, even though -3 is also part of the \(x \geq -1\) inequality.

Remember, the union considers all values that can be drawn from each inequality. Any value making either inequality true becomes part of our solution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators (like +, -, ×, ÷). They form the backbone of solving equations and inequalities, as they describe the relationships we are analyzing. In the exercise, expressions like \(x \geq -1\) and \(x = -3\) are examples given in inequality form.

• These expressions each describe specific conditions for \(x\).
• We assess them individually or together to find particular values of \(x\).

Understanding algebraic expressions is key to unlocking and working with various inequalities, making it an essential skill in mathematics.