1. What is a Linear Inequality?
A linear inequality is similar to a linear equation, but instead of an equals sign, it uses inequality signs: \( < \) (less than), \( > \) (greater than), \( \leq \) (less than or equal to), or \( \geq \) (greater than or equal to).
For example:
- \( y < 2x + 1 \)
- \( 3x - y \geq 7 \)
2. Graphing Linear Inequalities
When graphing linear inequalities, you'll plot a line (just like a linear equation), but you'll also show which side of the line your solution set lies on.
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Treat the inequality as an equation and graph the line:
- If your inequality is \( < \) or \( > \), draw a dashed line (because the line itself isn't part of the solution).
- If your inequality is \( \leq \) or \( \geq \), draw a solid line (because the line itself is part of the solution).
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Shade the correct side of the line:
- Pick a test point (not on the line). \( (0,0) \) is a common choice unless the line passes through the origin.
- If the test point satisfies the inequality, shade that side of the line. If not, shade the opposite side.
Example:
Graph the inequality \( y > 2x + 1 \).
- Draw the line \( y = 2x + 1 \) as a dashed line because of the \( > \) sign.
- Test the point \( (0,0) \).
- Is \( 0 > 2(0) + 1 \)?
- Is \( 0 > 1 \)? No, it's not.
3. Solving Linear Inequalities
The process of solving linear inequalities is similar to solving linear equations. However, there's one key difference:
When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality.
Example:
Solve the inequality: \( -2x > 8 \).
- Divide each side by -2 (and flip the inequality!): \( x < -4 \)
The solution is \( x < -4 \).
4. Compound Inequalities
Sometimes, you might encounter two inequalities that are linked by "and" or "or".
- "And" means both conditions must be true.
- "Or" means at least one of the conditions must be true.
Example:
Solve \( x > 2 \) and \( x < 5 \).
Graphically, this means \( x \) is greater than 2 and less than 5. So, the solution is all numbers between 2 and 5. It can be represented as: \( 2 < x < 5 \).
Final Thoughts
Remember to practice solving and graphing linear inequalities to reinforce your understanding. Over time, with practice, you'll become more comfortable working with them.
Example:
Solve and graph the inequality \(2x + 6 > 12\).
Solution:
Step 1: Isolate the variable \(x\).
Starting with: \[2x + 6 > 12\]
Subtract 6 from each side: \[2x > 6\]
Now, divide both sides by 2: \[x > 3\]
Step 2: Graph the inequality.
- Draw a number line.
- Place an open circle on 3 (since the inequality is strictly \(>\), meaning 3 is not included in the solution).
- Shade all the values to the right of 3 because they are greater than 3.
The graphical representation shows that \(x\) is any number greater than 3.
Final Answer:
\[x > 3\]
This means that any number greater than 3 is a solution to the inequality.
Solving Linear Inequalities
Linear inequalities are similar to linear equations, but instead of having an equal sign \((=)\), they have inequality signs such as \(<, >, \leq,\) or \( \geq \).
For example:
\[ 2x - 3 > 7 \]
To solve this inequality, follow these steps:
- Treat it similarly to an equation, with the aim to isolate \( x \).
- If you multiply or divide both sides by a negative number, remember to reverse the inequality sign.
Solution
Given the inequality:
\[ 2x - 3 > 7 \]
To solve for \( x \):
- Add 3 to both sides to get:
- Then, divide both sides by 2 to find the solution:
\[ 2x > 10 \]
\[ x > 5 \]
Thus, the solution to the inequality is \( x > 5 \).
System of Linear Inequalities
A system of linear inequalities consists of two or more linear inequalities with the same variables.
Example:
Consider the system of linear inequalities:
- \( 2x + y \leq 8 \) ……(i)
- \( x - 2y > 2 \) ……(ii)
Solution:
Step 1: Solve each inequality for \( y \).
From (i): \( y \leq -2x + 8 \)
From (ii): \( y < 0.5x - 1 \)
Step 2: Graph each inequality on a coordinate plane.
For (i):
- The line \( 2x + y = 8 \) is plotted.
- Since it's "≤", we will shade the region below the line (including the line).
For (ii):
- The line \( x - 2y = 2 \) or \( y = 0.5x - 1 \) is plotted.
- Since it's ">", we will shade the region above the line (not including the line).
Step 3: The solution to the system will be the overlapping shaded region.
