Understanding Inequalities in Algebra
Inequalities are mathematical statements that compare two expressions using symbols such as <, >, ≤, or ≥. They are fundamental in algebra and are used to describe ranges of possible values rather than exact values.
An inequality is a statement that shows the relationship between two expressions that are not necessarily equal.
Common symbols:
- < : less than
- > : greater than
- ≤ : less than or equal to
- ≥ : greater than or equal to
To solve a linear inequality, perform similar steps as solving an equation, but remember to reverse the inequality sign when multiplying or dividing both sides by a negative number.
Example:
- Solve: \( 2x - 3 < 7 \)
- Add 3: \( 2x < 10 \)
- Divide by 2: \( x < 5 \)
A compound inequality combines two inequalities, often using 'and' or 'or'.
Example:
- \( 1 < x \leq 4 \) means x is greater than 1 and less than or equal to 4.
To solve quadratic inequalities, first solve the related equation, then test intervals between the solutions.
Example:
- Solve: \( x^2 - 4 > 0 \)
- Find roots: \( x = -2, 2 \)
- Test intervals: \( (-\infty, -2) \) and \( (2, \infty) \) satisfy the inequality.
Absolute value inequalities involve expressions like \( |x| < a \) or \( |x| > a \).
Rules:
- \( |x| < a \) means \( -a < x < a \)
- \( |x| > a \) means \( x < -a \) or \( x > a \)
Inequalities can be represented on a number line or coordinate plane. Use open circles for < and >, and closed circles for ≤ and ≥.
For two variables, the solution is a region of the plane.
Inequalities are used in real-world problems such as budgeting, engineering constraints, and optimization.
They help describe ranges of possible solutions rather than a single answer.
- Solve: \( 3x + 5 > 11 \)
- Solve: \( 2x - 7 \leq 9 \)
- Solve and graph: \( |x - 2| < 5 \)
- Solve: \( x^2 - 1 \geq 0 \)