Inequalities in Mathematics

Introduction

Inequalities are mathematical statements that express the relationship of non-equality between two expressions. They play a significant role in algebra, calculus, and optimization problems.

Common symbols used in inequalities include $>$, $<$, $\geq$, and $\leq$.

Types of Inequalities

Strict Inequalities

Strict inequalities indicate that one quantity is either strictly greater than or strictly less than another:

• $a > b$: a is greater than b.
• $a < b$: a is less than b.

Weak Inequalities

Weak inequalities include equality along with greater than or less than:

• $a \geq b$: a is greater than or equal to b.
• $a \leq b$: a is less than or equal to b.

Compound Inequalities

These involve two inequalities joined by "and" or "or":

Properties of Inequalities

• Transitive Property: If $a > b$ and $b > c$, then $a > c$.
• Additive Property: Adding the same value to both sides does not change the inequality:
  $$a > b \implies a + c > b + c$$
• Multiplicative Property: Multiplying by a positive number preserves the inequality, but multiplying by a negative number reverses it:
  $$a > b \implies ac > bc \text{ if } c > 0$$
  $$a > b \implies ac < bc \text{ if } c < 0$$

Methods to Solve Inequalities

Linear Inequalities

For inequalities like $ax + b > 0$, isolate $x$:

1. Subtract $b$ from both sides.
2. Divide by $a$, reversing the sign if $a < 0$.

Quadratic Inequalities

Solve quadratic inequalities by:

1. Factoring the quadratic expression.
2. Determining the intervals where the expression is positive or negative.

For example, solve $x^2 - 4 > 0$:

The solution is $x < -2$ or $x > 2$.

Absolute Value Inequalities

Inequalities involving absolute values can be split into two cases:

Graphical Representation

Inequalities can be visualized on a number line or coordinate plane:

• Number Line: Shade the region that satisfies the inequality.
• Coordinate Plane: Shade the appropriate region of the plane for two-variable inequalities like $y > 2x + 3$.

Applications of Inequalities

Inequalities have practical applications in various fields:

• Economics: Budget constraints and optimization problems.
• Physics: Analyzing motion and forces within limits.
• Engineering: Ensuring safety margins in designs.
• Statistics: Confidence intervals and probability bounds.

Example

A factory produces widgets at a cost of $5x$ dollars, where $x$ is the number of widgets. If the budget is limited to 500 dollars, the inequality is:

Solving gives $x \leq 100$.

Practice Questions with Solutions

Basic Level Questions

1. Solve: $3x + 5 > 11$

   Solution:

   $$3x + 5 > 11$$
   $$3x > 6$$
   $$x > 2$$

   The solution is $x > 2$.

2. Solve: $|x - 3| \leq 4$

   Solution:

   $$-4 \leq x - 3 \leq 4$$
   $$-1 \leq x \leq 7$$

   The solution is $-1 \leq x \leq 7$.

Intermediate Level Questions

3. Solve: $2x^2 - 3x - 2 \geq 0$

   Solution:

   $$(2x + 1)(x - 2) \geq 0$$

   The critical points are $x = -\\frac{1}{2}$ and $x = 2$. Testing intervals:

   • $x < -\\frac{1}{2}$: Positive
   • $-\\frac{1}{2} < x < 2$: Negative
   • $x > 2$: Positive

   The solution is $x \leq -\\frac{1}{2} \text{ or } x \geq 2$.

4. Solve: $x^2 + 4x + 3 < 0$

   Solution:

   $$(x + 1)(x + 3) < 0$$

   The critical points are $x = -1$ and $x = -3$. Testing intervals:

   • $x < -3$: Positive
   • $-3 < x < -1$: Negative
   • $x > -1$: Positive

   The solution is $-3 < x < -1$.

Advanced Level Questions

5. Solve: $3^{x+1} \geq 27$

   Solution:

   $$3^{x+1} \geq 3^3$$
   $$x + 1 \geq 3$$
   $$x \geq 2$$

   The solution is $x \geq 2$.

6. Solve: $2x^3 - 3x^2 - 2x + 3 \leq 0$

   Solution:

   Factorizing:

   $$(x - 1)(2x^2 - x - 3) \leq 0$$

   The quadratic factor $2x^2 - x - 3$ gives roots $x = -1$ and $x = \\frac{3}{2}$.

   Critical points are $x = -1, x = \\frac{3}{2}, x = 1$. Testing intervals:

   The solution is $-1 \leq x \leq \\frac{3}{2}$.

Conclusion

Inequalities form a foundational concept in mathematics with wide-ranging applications. Practice solving different types of inequalities to build a strong mathematical base.