Linear inequalities play a pivotal role in algebra, representing situations where exact equality isn't the case. Unlike linear equations that use the equals sign, linear inequalities use symbols like <, >, ≤, or ≥. These symbols mean "less than", "greater than", "less than or equal to", and "greater than or equal to", respectively. The objective when solving these inequalities is to determine the range of values that make the inequality hold true.
Introduction to Linear Inequalities
Linear inequalities bear a resemblance to linear equations, but there's a key distinction: while equations yield a definite solution, inequalities offer a range of solutions. For instance, solving the equation x + 3 = 5 would give x = 2. However, for the inequality x + 3 > 5, the solution would be x > 2, implying any number greater than 2 would make the inequality true.
Techniques for Solving Linear Inequalities
1. Isolate the Variable: The primary aim when tackling linear inequalities is to have the variable on one side. This can be achieved by:
- Adding or subtracting terms on both sides to group like terms.
- Multiplying or dividing both sides by the same non-zero number to single out the variable. Crucially, if you multiply or divide by a negative number, the inequality symbol must be reversed.
2. Graphing Solutions: After solving the inequality, its solution can be depicted graphically on a number line:
- Open circles are used for < and > symbols, indicating the number isn't part of the solution set.
- Closed circles are for ≤ and ≥ symbols, showing the number is included in the solution set.
3. Compound Inequalities: These inequalities have more than one inequality symbol. They can be:
- AND inequalities (e.g., 1 < x ≤ 5): Both conditions must hold true.
- OR inequalities (e.g., x < 1 or x > 5): At least one of the conditions must be true.
Properties of Linear Inequalities
Understanding the properties of inequalities is vital for accurate solutions:
1. Addition and Subtraction: If you add or subtract the same number from both sides of an inequality, the inequality remains unchanged. For instance, if a > b, then a + c > b + c for any real number c.
2. Multiplication and Division: Multiplying or dividing both sides of an inequality by a positive number keeps the inequality unchanged. However, if done by a negative number, the direction of the inequality must be reversed. For example, if a > b and c < 0, then ac < bc.
3. Combining Inequalities: If a > b and c > d, then a + c > b + d.
4. Inverse Property: If a > b, then -a < -b.
Example Questions
Example 1: Solve the inequality 2x + 3 > 7.
Solution:
1. Subtract 3 from both sides: 2x > 4.
2. Divide both sides by 2: x > 2.
Example 2: Solve the compound inequality 1 ≤ x + 3 < 5.
Solution:
- 1. Subtract 3 from all parts of the inequality: -2 ≤ x < 2.
Example 3: Solve the inequality 3 - x ≤ 2x + 1.
Solution:
1. Add x to both sides: 3 ≤ 3x + 1.
2. Subtract 1 from both sides: 2 ≤ 3x.
3. Divide both sides by 3: 2/3 ≤ x or x ≥ 2/3.
Example 4: Solve the compound inequality x - 4 > 2 or 2x + 1 < 5.
Solution:
1. For the first inequality, add 4 to both sides: x > 6.
2. For the second inequality, subtract 1 from both sides: 2x < 4. Then, divide by 2: x < 2.
FAQ
To represent the solution of an inequality on a number line, you'll use dots and shading. For strict inequalities (< or >), use an open dot to show that the endpoint is not included in the solution. For non-strict inequalities (≤ or ≥), use a solid dot to indicate that the endpoint is part of the solution. After marking the endpoint, shade the number line to the left for "less than" inequalities and to the right for "greater than" inequalities. The shaded region represents all the values that satisfy the inequality.
Yes, you can have compound inequalities that combine both "AND" and "OR" conditions, although they might be more complex to solve and interpret. An "AND" condition means both inequalities must be true simultaneously, while an "OR" condition means at least one of the inequalities must be true. When solving such compound inequalities, it's helpful to solve each inequality separately first and then combine the solutions based on the given conditions. Graphically, "AND" conditions often result in a single continuous shaded region on the number line, while "OR" conditions can result in two or more separate shaded regions.
When solving inequalities involving absolute values, it's essential to remember the definition of absolute value: the distance of a number from zero on the number line. For example, to solve |x - 2| > 5, you're looking for values of x that are more than 5 units away from 2. This leads to two inequalities: x - 2 > 5 (which gives x > 7) and x - 2 < -5 (which gives x < -3). So, the solution would be x < -3 or x > 7. Always break down absolute value inequalities into two separate inequalities to solve them.
When you multiply or divide both sides of an inequality by a negative number, the order of the numbers gets reversed. For example, consider the simple fact that 2 is greater than 1. If you multiply both sides by -1, you get -2 and -1. Now, -2 is less than -1, not greater. So, the inequality sign must be reversed to maintain the relationship's truth. Failing to reverse the inequality sign when multiplying or dividing by a negative number can lead to incorrect solutions.
A strict inequality is one that uses the symbols < (less than) or > (greater than). It indicates that one value is strictly less than or strictly greater than another, without being equal to it. For example, x > 3 means that x can be any value greater than 3, but not 3 itself. On the other hand, a non-strict inequality uses the symbols ≤ (less than or equal to) or ≥ (greater than or equal to). It means that one value can be less than, equal to, or greater than another value. For instance, x ≤ 3 means x can be any value less than or equal to 3, including 3 itself.
Practice Questions
To solve the inequality 5x - 3 <= 2x + 6, we'll first get all the terms with x on one side and the constants on the other side.
Subtracting 2x from both sides, we get: 3x - 3 <= 6
Next, adding 3 to both sides, we have: 3x <= 9
Finally, dividing both sides by 3, we get: x <= 3
This means that the solution to the inequality is all values of x that are less than or equal to 3. On a number line, this would be represented by a solid dot at 3 (indicating that 3 is included in the solution) and shading to the left of 3.
For the first inequality x + 4 > 7: Subtracting 4 from both sides, we get: x > 3
For the second inequality 3x - 5 < 4: Adding 5 to both sides, we have: 3x < 9
Dividing both sides by 3, we get: x < 3
So, the solution to the compound inequality is x > 3 OR x < 3. This means that all values of x that are either greater than 3 or less than 3 are solutions to the compound inequality. On a number line, this would be represented by an open dot at 3 (indicating that 3 is not included in the solution) with shading to the right of 3 for x > 3 and shading to the left of 3 for x < 3.
