Inequalities

A mathematical statement that compares two expressions and indicates which is larger, smaller, or possibly equal to the other is called an inequality. This term is used to evaluate the numerical value or magnitude of two expressions. In contrast to equations, which require precise equality, inequalities show a range of potential values. They define constraints or conditions.

Let's understand the practical application of inequalities using an example.  

Anthony is the younger brother of Lily. Lily's age is 13 years. So, what is Anthony's age? Here, no information about Anthony's age is given, but it does mention the relation with his sister's age. Since he is younger than Lily, it can be said that Anthony's age is less than 13. Thus, it can be written as Anthony's age < 13. 

Inequalities play an important role in mathematics, as they compare quantities which are not necessarily equal. Since their main function is to compare quantities, different symbols are used to help compare values. The different symbols are as follows:

The following table gives the detailed information about different symbols of inequalities:

Inequalities can be solved using specific rules. The following is a discussion of some of these rules:

Rule 1: The transitive property states that for three numbers, a, b, and c, the following rules apply: 

• a > c if a > b and b > c.
• If b < c and a < b, then a < c.
• If b ≥ c and a ≥ b, then a ≥ c.
• a ≤ c if a ≤ b and b ≤ c.

Rule 2: Switching the LHS and RHS of the expressions causes the inequality to reverse. We refer to it as a symmetric property.

• b < a if a > b.
• If a < b, then b > a.
• b ≤ a if a ≥ b.
• b ≥ a if a ≤ b.

Rule 3: Adding or subtracting the same constant, k, from both sides of an inequality doesn’t affect the inequality.
 

• If a > b, then a + k > b + k.
• If a > b, then a - k > b - k.

The same is true for other inequalities as well.

• If a < b, then a + k < b + k.
• If a < b, then a - k < b - k.
• If a ≤ b, then a + k ≤ b + k.
• If a ≤ b, then a - k ≤ b - k.
• If a ≥ b, then a + k ≥ b + k.
• If a ≥ b, then a - k ≥ b - k.

Rule 4: Multiplying or dividing both sides by a positive constant k does not affect the inequality.  
 

• If a > b, then ak > bk
• If a < b, then ak < bk
• If a ≤ b, then ak ≤ bk
• If a ≥ b, then ak ≥ bk

But, multiplying or dividing both sides by a negative constant k reverses the direction of inequality. This change happens due to change in the order of numbers on the number line.
 

• If a > b, then ak < bk
• If a > b, then ak < bk
• If a ≥ b, then ak ≤ bk
• If a ≤ b, then ak ≥ bk

Rule 5: Taking the reciprocal of numbers, flips the inequality sign
 

• If, a < b, then 1/a > 1/a
• If a > b, then 1/a < 1/b

This happens due to change in their sizes, when taking reciprocal.

For example, cutting a pizza in 5 pieces gives you smaller slices, than getting a slice of pizza that is divided in only 2 parts. That is why, 2 > 5, but ½ < ⅕.
 

Rule 6: The square of a given number is always greater than or equal to zero: a² ≥ 0

Since, the square of any numbers gives positive values, hence, they are always greater or equal to 0. 

Example, (-1)²  = 1, (1)² = 1, 0² = 0

Rule 7: Taking squares or square roots on both sides does not change the inequality sign. 
 

• If a > b, then a² > b²
• If a < b, then a² < b²
• If a > b, then √a > √b²
• If a > b, then √a > √b

Example: As we know, 4 < 9, taking square root on both side gives, √4 < √9, which is true. 
 

The following table summarizes, the above rules according to the change in sign of inequalities:

 

This is an organized method for solving inequalities and accurately identifying the solution set.

Step 1: First, simplify the inequality to isolate the variable.

Step 2: Solve the equation to find one or more values.

Step 3: Write each value on the number line.

Step 4: In addition, use open circles to symbolize all excluded values on the number line.

Step 5: Determine the intervals.

Step 6: Choose a random number from each interval and enter it into the inequality to see if it is satisfied.

Step 7: The intervals that are satisfied are the solutions.

However, we typically use algebraic operations like addition, subtraction, multiplication, and division to solve simple inequalities (linear). 

The following considerations must be made when writing the interval notation solution to an inequality.

1. Use the closed brackets '[' or ']' if the endpoint is included, as in ≤ or ≥.
    
2. If the endpoint is omitted, as in the case of < or >, use the open brackets "(" or ")".
    
3. Use '(' for -∞ and ')' for ∞ to indicate endpoints are not included.

Now, let's understand these interval notations using examples:
 

Plotting the ‘equals’ line and shading the relevant region are the first steps in graphing inequalities involving two variables. Four steps are involved:

1. Place y on the equation's left side and all other values on the right.
    
2. Plot the y = line, displaying a solid line for y ≤ or y ≥ and a dashed line for y < or y >.
    
3. For a less-than inequality (y < or y ≤), shade below the line.
    
4. For a greater-than inequality (y > or y ≥), shade above the line.

How to Solve Polynomial Inequalities?

Finding the values of a variable that satisfy an inequality when the expression is a polynomial or a rational function is known as solving polynomial inequalities.

Linear inequalities (including one-step and two-step inequalities), compound inequalities, quadratic inequalities, absolute value inequalities, and rational inequalities will all be covered in this response, along with concise instructions, examples, and answers provided in interval and set notation.

1. Linear Inequalities

Polynomials of degree 1 (example, 𝑎𝑥 + 𝑏 < 𝑐) are involved in linear inequalities. Like linear equations, they are solved by isolating the variable while paying attention to inequality signs.

a. One-Step Inequalities

These can be solved using just one operation to isolate the variable. 

Let’s consider the inequality \(𝑥 + 3 > 7\).

• To separate x from the equation, subtract 3 from both sides: \(x > 4\).
   
• In set notation, the solution is \(\{x ∈ R ∣ x > 4\}\).
   
• In interval notation, it’s written as (4, ∞) which means all real numbers greater than 4.
   
• On a number line, this is expressed with the help of an open circle at 4 and an arrow pointing right, as 4 is not included.

This method ensures the inequality keeps pointing in the correct direction and isolates the variable easily.

b. Two-Step Linear Inequalities

Let’s look at this with the help of an example. Consider the equation \(2x − 5 ≤ 3\).

• Adding 5 to both sides → \(2x ≤ 8\).
   
• Dividing by 2 → \(x ≤ 4\).
   
• In set notation, the solution is \(\{𝑥 ∈ 𝑅 ∣ 𝑥 ≤ 4\}\)
   
• In interval notation, the solution is (−∞, 4), meaning all numbers less than or equal to 4. 
   
• On a number line, this is shown with the help of a closed circle at 4 and shading towards the left as 4 is included.

2. Compound Inequalities

Compound inequalities combine two inequalities with 'and' (intersection, where both conditions must hold) or 'or' (union, where at least one condition holds). Solve each inequality independently, then combine the outcomes according to the connector. 

1. Take the "and" compound inequality\( −2 < 𝑥 ≤ 3\) as an example.
    
2. Since \(x > −2\) and \(x ≤ 3\), the intersection of these sets—that is, all numbers between -2 and 3, including 3 — is the answer.
    
3. This is represented in interval notation as (−2, 3) and in set notation as \(\{x ∈ R ∣ − 2 < x ≤ 3\}\).
    
4. Draw a closed circle at  \(𝑥=3 \) and an open circle at \(𝑥=−2\) on a number line, then shade the area in between.
    
5. The solution for an "or" example such as \(𝑥<−1\) or \(𝑥>2\) is \((−∞, −1) ∪ (2, ∞)\), graphed with open circles at -1 and 2, shading left of -1 and right of 2.

This method captures combined constraints well.

3. Inequalities that are quadratic

Analysis is necessary to determine whether the quadratic expression is positive, negative, or zero in quadratic inequalities, which involve second-degree polynomials.

To determine whether the inequality holds, rewrite the inequality with zero on one side, solve the corresponding equation to determine the roots, divide the number line into intervals using these roots, and test points in each interval. 

1. To solve \(𝑥² − 𝑥 − 6 > 0\), for instance, factor the equation \(x² − x − 6 = 0\) as \((x − 3)(x + 2) = 0\), yielding roots \(x = 3\) and \(𝑥 = −2\).
    
2. \((−∞, −2), (−2, 3), \text {and } (3, ∞)\) are the intervals.
    
3. Testing points: the expression is positive at \(x=−3\), negative at x=0, and positive at \(x=4\).
    
4. The answer is therefore \(x < −2 \ or \ x > 3\), or in interval notation \((−∞, −2) ∪ (3, ∞)\), and in set notation \(\{x ∈ R ∣ x < −2 \ or \ x > 3\}\).
    
5. On a number line with open circles at -2 and 3 and shading outward, the parabola \(𝑦 = 𝑥² − 𝑥 − 6\) is graphically located above the x-axis to the right of \(𝑥 = 3\) and to the left of \(𝑥 = −2\).

This technique uses the behavior of the quadratic to identify areas of the solution.
 

4. Inequalities of Absolute Value

Expressions of the form \(∣𝑎𝑥 + 𝑏∣ < 𝑐\) or \(∣𝑎𝑥 + 𝑏∣ > 𝑐\) are examples of absolute value inequalities. These are solved by transforming them into compound inequalities according to the definition of the absolute value. For \(∣𝑎𝑥 + 𝑏 ∣<𝑐\), rewrite as \(−𝑐 < 𝑎𝑥 + 𝑏 < 𝑐\); for \(∣𝑎𝑥 + 𝑏 ∣> 𝑐\), rewrite as \(𝑎𝑥 + 𝑏 <−𝑐\) or \(𝑎𝑥 + 𝑏 > 𝑐\).

Consider \(∣2 𝑥 − 1∣ ≤ 5\).

1. Rewrite as follows: \(−5 ≤ 2 𝑥 − 1 ≤ 5\).
    
2. Add 1 to get \(−4 ≤ 2 𝑥 ≤ 6\),
    
3. Divide by 2 to get \(−2 ≤ 𝑥 ≤ 3\).
    
4. The answer is \(\{𝑥 ∈ 𝑅∣− 2 ≤ 𝑥 ≤ 3\}\) in set notation and \([−2, 3]\) in interval notation. This is represented graphically as a number line with closed circles shaded between 𝑥 = −2 and 𝑥 = 3.
    
5. For \(∣𝑥 + 1 ∣ > 2\), rewrite as \(𝑥 + 1 < −2\) or \(𝑥 + 1 > 2\), resulting in \(𝑥 < −3\) or \(𝑥 > 1\), or \((−∞, −3) ∪ (1, ∞)\), graphed with open circles at -3 and 1 shading outward.

Absolute value problems are reduced to manageable linear inequalities using this method.

5. Rational Inequalities

Ratios of polynomials, like \(𝑝(𝑥)\ 𝑞(𝑥) > 0\), are the subject of rational inequalities, which call for determining whether the expression is positive, negative, or zero while avoiding points where the denominator is zero. Divide the number line into intervals, test points within each interval, move all terms to one side, and determine the denominator and numerator zeros. 

For instance, figure out \(\frac{(x + 1)}{(x − 2)} ≥ 0\).

1. The critical points are \(𝑥 = −1\) (numerator zero) and \(𝑥 = 2\) (denominator zero).
    
2. The intervals are \((−∞, −1), (−1, 2), \ and \ (2, ∞)\). Test points: the expression is positive at \(𝑥 = −2\), negative at \(x = 0\), and positive at \(x = 3\).
    
3. Since ≥, include \(x = -1\), but leave out \(x = 2\).
    
4. The answer is \(\{x ∈ R∣ x ≤ −1 \ or\ x > 2\} \) in set notation and \((−∞, −1] ∪ (2, ∞)\) in interval notation.
    
5. A number line is represented graphically by a closed circle at \(𝑥 = −1\), an open circle at \(𝑥 = 2\), and shading to the left of -1 and to the right of 2.
    
6. The function's graph has a vertical asymptote at \(𝑥 = 2\) and is non-negative in the solution regions.

Discontinuities in rational expressions are handled with care in this method.

Students can find inequalities a bit confusing in the beginning. Here are a few tips and tricks to help students understand inequalities easily.
 

1. Use a number line to represent numbers and inequalities.
    
2. Practice rules of inequalities by solving problems to memorize them correctly.
    
3. Relates inequalities to real life situations for better understanding.
    
4. Practice using inequality symbols by applying to real-life examples. Example, 2 cookies < 4 cookies.
    
5. Carefully use interval notations.

Real-world applications of inequalities include comparing values, establishing boundaries, and making decisions in domains such as daily planning, business, and budgeting.

1. Finance and Budgeting: Inequalities prevent you from going over your income when you plan your monthly expenses.
   
   Assume you have to set aside $3,000 for utilities (z), food (y), and rent (x) to make sure your overall spending stays within your income. The inequality is: \(x + y + z ≤ 3000\). It makes sure you don't spend more than you make. 
   
   This same concept can be used by students to determine how much to save, spend on entertainment, or put aside for emergencies, always keeping overall spending within their allocated budget.
   
    
2. Engineering and Design: Inequalities are used by engineers to make sure that structures meet safety regulations.
   
   For example, in order to prevent collapse, a bridge's load capacity must satisfy \(𝑊 ≤ 𝐶\), where 𝐶 is the maximum capacity and 𝑊 is the weight it bears. People on a bridge must weigh not more than 7,000 kg overall if the bridge can support 10,000 kg and the materials weigh 3,000 kg.
   
   Engineers prevent collapse and safeguard public safety by implementing these inequalities at every stage of the design process.
   
    
3. Medicine and healthcare: Inequality determines safe drug dosages.
   
   For example, a doctor might prescribe a drug whose dosage 𝐷 must be \(5 ≤ 𝐷 ≤ 20\) mg per kg, where 𝐷 represents the dosage per kilogram of body weight. This amounts to \(250 ≤ 𝐷 ≤ 1000\) mg for a patient weighing 50 kg.
   
   In order to maintain the drug's effectiveness without reaching toxic levels and guarantee favorable patient outcomes, medical students learn to compute these bounds. 
   
    
4. Business and Inventory Management: To maximize stock levels, retailers employ inequalities.
   
   A store might, for instance, keep inventory 𝐼 such than \(100 ≤ 𝐼 ≤ 500\). This means that you should never order so few that the shelves are empty (less than 100) or overstock (more than 500).
   
   A reorder is initiated when sales spike and 𝐼 moves toward the lower bound; further orders are stopped when sales slow and 𝐼 moves toward the upper bound. Business students observe how these disparities power automated ordering systems and stop waste or lost sales.
   
    
5. Logistics and Transportation: Routes and schedules are optimized by inequalities in transportation and logistics.
   
   To make sure the load doesn't go over the limit, a delivery truck with a 2,000 lbs capacity might use \(𝑊_1 + 𝑊_2 +… + 𝑊_n ≤ 2000\), where 𝑊_n is the weight of each package. A fourth package can weigh no more than 250 pounds if the combined weight of the three packages — 600, 850, and 300 pounds — is 1,750 pounds.
   
   This straightforward inequality maximizes delivery efficiency, prevents fines, and keeps trucks within safe bounds. Students observe how these regulations support route planning and assist businesses in cutting fuel expenses.