write notes on solution sets of equations and inequalities

Answer 1

Solution sets of equations and inequalities refer to the collection of values or values ranges that satisfy the given equation or inequality. Here are some important points to note about solution sets:

1. Equations: The solution set of an equation is the set of values that makes the equation true. For example, in the equation 3x + 5 = 14, the solution is x = 3, as when x is substituted in the equation, it holds true.

2. Inequalities: The solution set of an inequality consists of all values that satisfy the given inequality. For example, in the inequality 2x + 3 > 10, the solution set is x > 3. This means that any value of x greater than 3 will make the inequality true.

3. Solution sets for equations can be a single value, multiple values, or no solution. For instance, the equation x^2 = 4 has two solutions, x = 2 and x = -2, whereas the equation x + 3 = 10 has a single solution x = 7. However, the equation 2x + 1 = 2x + 3 doesn't have a solution, as the equation is contradictory (both sides are never equal).

4. Solution sets for inequalities can be expressed using inequalities or intervals. For example, the inequality 2x - 3 ≤ 7 can be represented as x ≤ 5 by subtracting 2x from both sides. Alternatively, it can be expressed as the closed interval [-∞, 5].

5. Solution sets of systems of equations refer to the values that satisfy all the given equations simultaneously. For example, in the system of equations {2x + y = 7, x - y = 1}, the solution set is x = 2 and y = 3, as these values solve both equations simultaneously.

6. When graphing solution sets on a number line or coordinate plane, open circles are used to represent values not included in the solution set, while closed circles indicate values that are included.

7. Solution sets can also be classified as infinite or finite. For example, the solution set of the equation x + 2 = 10 is finite, containing a single value x = 8. However, the solution set of the inequality x > 3 is infinite, as any value of x greater than 3 will satisfy the inequality.

Remember to always verify the solution set by substituting the values back into the original equation or inequality to ensure they hold true.