2-3-1 Linear Inequalities Explained
Key Concepts of Linear Inequalities
Linear inequalities are mathematical statements that compare two expressions using inequality symbols (<, >, ≤, ≥). They are similar to linear equations but involve inequalities rather than equalities.
1. Inequality Symbols
The four main inequality symbols are:
- < (less than)
- > (greater than)
- ≤ (less than or equal to)
- ≥ (greater than or equal to)
Example:
\[ 3x + 2 < 10 \]
This inequality states that the expression \( 3x + 2 \) is less than 10.
2. Solving Linear Inequalities
Solving linear inequalities involves finding the values of the variable that make the inequality true. The process is similar to solving linear equations, but with an important distinction: when multiplying or dividing by a negative number, the inequality symbol must be reversed.
Example:
Solve the inequality \( -2x + 5 > 1 \):
1. Subtract 5 from both sides: \( -2x > -4 \)
2. Divide both sides by -2 (reverse the inequality): \( x < 2 \)
The solution is \( x < 2 \).
3. Graphing Linear Inequalities
Graphing linear inequalities on a number line helps visualize the solution set. For inequalities involving ≤ or ≥, use a closed circle (●) to indicate the endpoint is included. For inequalities involving < or >, use an open circle (○) to indicate the endpoint is not included.
Example:
Graph the inequality \( x ≥ -3 \):
1. Draw a number line.
2. Place a closed circle at -3.
3. Shade to the right of -3 to indicate all values greater than or equal to -3.
4. Compound Inequalities
Compound inequalities involve two inequality statements joined by "and" or "or." The solution set is the intersection (and) or union (or) of the individual solution sets.
Example:
Solve the compound inequality \( -1 ≤ 2x + 3 < 7 \):
1. Subtract 3 from all parts: \( -4 ≤ 2x < 4 \)
2. Divide all parts by 2: \( -2 ≤ x < 2 \)
The solution is \( -2 ≤ x < 2 \).
Practical Applications
Linear inequalities are used in various real-world applications, such as setting budget constraints, determining time limits, and solving optimization problems. Understanding linear inequalities helps in making informed decisions based on mathematical constraints.