what is the main difference you have to keep in mind when solving an equation v. an inequality?
When solving an equation, the goal is to find the value(s) that satisfy the equation and make it true. Equations are primarily concerned with equality.
On the other hand, when solving an inequality, the goal is to find the range of values that satisfy the inequality. Inequalities involve a comparison of values, typically using symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to).
The main difference to keep in mind when solving an inequality is how the solution is affected when performing operations on both sides of the inequality sign. There are three crucial aspects to consider:
1. Multiplication or Division by a negative number: If an inequality is multiplied or divided by a negative number, the direction of the inequality must be reversed. For example, if you have the inequality 2x > 6 and you multiply both sides by -1, you need to change the inequality sign to obtain -2x < -6.
2. Addition or Subtraction by a negative number: Adding or subtracting a negative number is equivalent to adding or subtracting the positive number of the same value. No changes are required to the direction of the inequality. For example, if you have the inequality x + 3 < 7 and you subtract 3 from both sides, you have x < 4.
3. Multiplication or Division by a positive number: When multiplying or dividing an inequality by a positive number, there is no need to change the direction of the inequality. For example, if you have the inequality 3x < 9 and you divide both sides by 3, you have x < 3.
It is important to note that when solving inequalities, the solution can often be expressed in different forms, such as interval notation or set notation, depending on the context and requirements of the problem.
When solving an equation, the goal is to find the value(s) of the variable that make the equation true. In contrast, when solving an inequality, the goal is to find the range of values of the variable that make the inequality true.
Here are three key differences to keep in mind when solving an equation versus an inequality:
1. Solution sets:
- For an equation, there can be one or more specific solutions.
- For an inequality, there can be a range of solutions or an infinite number of solutions.
2. Equality versus inequality:
- In an equation, the symbol used is typically an equal sign (=) to show that two expressions are equivalent.
- In an inequality, symbols like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) are used to compare the two sides of the inequality.
3. Operations on both sides:
- When solving an equation, various mathematical operations can be performed on both sides of the equal sign, as long as you do the same thing to both sides.
- When solving an inequality, different operations can be performed on both sides, but you must be careful about reversing the direction of the inequality when multiplying or dividing by a negative number.
It's important to note that the process of solving an inequality is similar to solving an equation, but you must consider the directionality of the inequality symbol and adjust accordingly.