Inequalities and absolute value
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1) Inequalities are relationships between expressions that are not equal, using symbols like ≠, >, or <. They can be solved using algebra by adding or subtracting the same quantity to both sides or multiplying/dividing both sides by a positive number. 2) Solving an inequality involves isolating the variable by performing inverse operations while maintaining the direction of the inequality symbol. For example, solving x + 3 > 2 by subtracting 3 from both sides results in x > -1. 3) Absolute value refers to the distance of a number from zero on the number line. It is represented by vertical bars around an expression, such as |x-y| being the distance between x and y.
Inequalities and absolute value
- 2. Inequalities The relation between two expressions that are not equal, employing a sign such as ≠ ‘not equal to’, > ‘greater than’, or < ‘less than’. Examples of inequalities : 5 < 7 ( 5 is less then 7 ) X < 5 ( X is less then 5 ) 2x + 3 > 0 ( 2x + 3 is greater then or equal to 0 )
- 3. Roles of solving Inequalities Inequalities are solved by using algebra and by using graphs. When solving an inequality : • you can add the same quantity to each side • you can subtract the same quantity from each side • you can multiply or divide each side by the same positive quantity If you multiply or divide each side by a negative quantity, the inequality symbol must be reversed.
- 4. Solving Inequalities Suppose we want to solve the inequality x + 3 > 2. We can solve this by subtracting 3 from both sides: x + 3 > 2 x > −1 So the solution is x > −1. This means that any value of x greater than −1 satisfies x + 3 > 2. Inequalities can be represented on a number line such as that shown in given Figure . The solid line shows the range of values that x can take. We put an open circle at −1 to show that although the solid line goes from −1, x cannot equal −1. A number line showing x > −1.
- 5. Example Suppose we wish to solve the inequality 4x + 6 > 3x + 7. First, we subtract 6 from both sides to give 4x > 3x + 1 Now we subtract 3x from both sides: x > 1 This is the solution. It can be represented on the number line as shown in Figure. A number line showing x > 1.
- 6. Absolute Value— Distance on the Number Line
- 7. What is absolute value? The absolute value of a number means the distance from 0. Examples : -5 is 5 units away from 0. So the absolute value of that -5 is 5. You cannot have negative distance, so it must be positive. The symbol for absolute value is two straight lines surrounding the number or expression for which you wish to indicate absolute value. |6| = 6 means the absolute value of 6 is 6.
- 8. What is a distance value? The absolute value is the distance between two values, depending on the expression contained within the absolute value sign. For example, the distance between two numbers x and y can be written as |x – y|. Therefore: |x| = |x – 0| is the distance of x from 0. Example : The absolute value of 2 + -7 is 5. Distance of sum from 0 : 5 units. The absolute value of 0 is 0. (This is why we don't say that the absolute value of a number is positive: Zero is neither negative nor positive.)
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