Solved Exercises on Introduction to Inequalities in Pre-algebra

Master introduction to inequalities: understanding inequality symbols, graphing solutions, and solving basic inequality problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Less than inequality
Exercise 1
Solve: \(x < 5\)
Graph the solution on a number line.
Definition:

Inequality: A mathematical statement that compares two expressions using symbols like \(<\), \(>\), \(\leq\), or \(\geq\).

Graphing method:
  1. Identify the inequality symbol
  2. Locate the boundary point on the number line
  3. Draw an open circle for \(<\) or \(>\) (not included)
  4. Draw a closed circle for \(\leq\) or \(\geq\) (included)
  5. Shade the region that satisfies the inequality
Inequality
\(x < 5\)
Boundary
\(5\)
Solution
All numbers less than 5
Step 1: Identify the inequality symbol

The symbol is \(<\) (less than), meaning values smaller than 5

Step 2: Locate the boundary point

The boundary point is 5

Step 3: Draw the boundary

Draw an open circle at 5 (since 5 is not included in the solution)

Step 4: Shade the solution region

Shade all numbers to the left of 5 (all numbers less than 5)

Solution: \(x \in (-\infty, 5)\)
Final answer:

All real numbers less than 5. The graph shows an open circle at 5 with shading to the left.

Applied rules:

• Open circle: For \(<\) or \(>\) (boundary not included)

• Shading direction: Less than means shade to the left

• Interval notation: \((-\infty, 5)\) represents all numbers less than 5

2 Greater than inequality
Exercise 2
Solve: \(y \geq 3\)
Graph the solution on a number line.
Definition:

Greater than or equal inequality: An inequality using \(\geq\) symbol, meaning the variable can be greater than or equal to the given value.

Inequality
\(y \geq 3\)
Boundary
\(3\)
Solution
All numbers greater than or equal to 3
Step 1: Identify the inequality symbol

The symbol is \(\geq\) (greater than or equal), meaning values 3 or larger

Step 2: Locate the boundary point

The boundary point is 3

Step 3: Draw the boundary

Draw a closed circle at 3 (since 3 is included in the solution)

Step 4: Shade the solution region

Shade all numbers to the right of 3 (all numbers greater than or equal to 3)

Solution: \(y \in [3, \infty)\)
Final answer:

All real numbers greater than or equal to 3. The graph shows a closed circle at 3 with shading to the right.

Applied rules:

• Closed circle: For \(\leq\) or \(\geq\) (boundary included)

• Shading direction: Greater than means shade to the right

• Interval notation: \([3, \infty)\) includes 3 and all numbers greater than 3

3 Less than or equal inequality
Exercise 3
Solve: \(z \leq -2\)
Graph the solution on a number line.
Definition:

Less than or equal inequality: An inequality using \(\leq\) symbol, meaning the variable can be less than or equal to the given value.

Inequality
\(z \leq -2\)
Boundary
\(-2\)
Solution
All numbers less than or equal to -2
Step 1: Identify the inequality symbol

The symbol is \(\leq\) (less than or equal), meaning values -2 or smaller

Step 2: Locate the boundary point

The boundary point is -2

Step 3: Draw the boundary

Draw a closed circle at -2 (since -2 is included in the solution)

Step 4: Shade the solution region

Shade all numbers to the left of -2 (all numbers less than or equal to -2)

Solution: \(z \in (-\infty, -2]\)
Final answer:

All real numbers less than or equal to -2. The graph shows a closed circle at -2 with shading to the left.

Applied rules:

• Closed circle: For \(\leq\) or \(\geq\) (boundary included)

• Shading direction: Less than means shade to the left

• Interval notation: \((-\infty, -2]\) includes -2 and all numbers less than -2

Rules and methods, laws,...
\(ax + b < c\)
Linear Inequality
Symbol
\(x < a\)
Open circle, shade left
Symbol
\(x > a\)
Open circle, shade right
Symbol
\(x \leq a\)
Closed circle, shade left
Symbol
\(x \geq a\)
Closed circle, shade right
Key definitions:

Inequality: A mathematical statement that compares two expressions using symbols like \(<\) (less than), \(>\) (greater than), \(\leq\) (less than or equal), or \(\geq\) (greater than or equal).

Solution Set: The set of all values that make the inequality true.

Boundary Point: The value where the inequality changes from true to false or vice versa.

Open Circle: Used on number lines for \(<\) or \(>\) to indicate the boundary point is not included in the solution.

Closed Circle: Used on number lines for \(\leq\) or \(\geq\) to indicate the boundary point is included in the solution.

Complete methodology:
  1. Identify the inequality symbol: Determine which comparison symbol is used
  2. Find the boundary point: Locate the critical value on the number line
  3. Determine inclusion: Decide if the boundary point is included in the solution
  4. Draw the circle: Open for \(<\) or \(>\), closed for \(\leq\) or \(\geq\)
  5. Shade the solution: Left for less than, right for greater than
  6. Express in interval notation: Use parentheses for excluded endpoints, brackets for included endpoints
Tip 1: Remember: Open circles for strict inequalities (< >), closed circles for inclusive inequalities (≤ ≥)
Tip 2: Less than shades to the left, greater than shades to the right
Tip 3: Test a value in the shaded region to verify your solution
Tip 4: Interval notation uses () for excluded values and [] for included values
Common errors: Confusing open/closed circles, shading in the wrong direction, misinterpreting inequality symbols, forgetting to flip the inequality when multiplying/dividing by negative numbers.
Memory aids: "Less than points left", "Greater than points right", "Equal sign means filled dot".
Solution: Exercises 4 to 5
4 Greater than inequality with negative number
Exercise 4
Solve: \(w > -4\)
Graph the solution on a number line.
Definition:

Strict inequality: An inequality using \(<\) or \(>\) symbols, where the boundary value is not included in the solution set.

Inequality
\(w > -4\)
Boundary
\(-4\)
Solution
All numbers greater than -4
Step 1: Identify the inequality symbol

The symbol is \(>\) (greater than), meaning values larger than -4

Step 2: Locate the boundary point

The boundary point is -4

Step 3: Draw the boundary

Draw an open circle at -4 (since -4 is not included in the solution)

Step 4: Shade the solution region

Shade all numbers to the right of -4 (all numbers greater than -4)

Step 5: Verify with a test value

Test \(w = 0\): \(0 > -4\) is true, confirming the right side is correct

Solution: \(w \in (-4, \infty)\)
Final answer:

All real numbers greater than -4. The graph shows an open circle at -4 with shading to the right.

Applied rules:

• Open circle: For \(>\) (boundary not included)

• Shading direction: Greater than means shade to the right

• Interval notation: \((-4, \infty)\) excludes -4 and includes all numbers greater than -4

5 Real-world application
Exercise 5
Sarah needs to save at least $20 for a concert ticket. Write an inequality for the amount she needs to save and graph the solution.
Definition:

Real-world inequality: Translating word problems into mathematical inequalities to represent constraints or conditions in practical situations.

Variable
Let \(s\) = amount saved
Inequality
\(s \geq 20\)
Solution
All amounts ≥ $20
Step 1: Define the variable

Let \(s\) = the amount Sarah needs to save

Step 2: Translate the condition

"At least $20" means $20 or more, so \(s \geq 20\)

Step 3: Graph the solution

Draw a closed circle at 20 (since 20 is included) and shade to the right

Step 4: Interpret the solution

Sarah needs to save $20 or more to buy the concert ticket

Inequality: \(s \geq 20\), Solution: \(s \in [20, \infty)\)
Final answer:

The inequality is \(s \geq 20\). Sarah needs to save at least $20. The graph shows a closed circle at 20 with shading to the right.

Applied rules:

• Word problem translation: "at least" means \(\geq\)

• Closed circle: For \(\geq\) (boundary included)

• Real-world context: Interpret the solution in the problem context

Comprehensive Summary: Introduction to Inequalities
\(ax + b < c, ax + b > c, ax + b \leq c, ax + b \geq c\)
Linear Inequality Forms
Key definitions:

Inequality: A mathematical statement that compares two expressions using inequality symbols: \(<\) (less than), \(>\) (greater than), \(\leq\) (less than or equal), \(\geq\) (greater than or equal).

Solution Set: The collection of all values that satisfy the inequality. Unlike equations that often have a single solution, inequalities typically have infinitely many solutions forming a range of values.

Boundary Point: The critical value where the inequality changes from true to false. This is the value that would make the inequality an equality.

Open Interval: An interval that does not include its endpoints, denoted with parentheses: \((a, b)\).

Closed Interval: An interval that includes its endpoints, denoted with brackets: \([a, b]\).

Complete methodology:
  1. Identify the inequality type: Determine which symbol is used and what it means
  2. Find the boundary point: Solve the corresponding equality to find the critical value
  3. Determine endpoint inclusion: Decide if the boundary is included based on the inequality symbol
  4. Graph on number line: Draw appropriate circle and shade the solution region
  5. Express solution: Write in interval notation or set-builder notation
  6. Verify: Test a value from the solution region to confirm correctness
Tip 1: The "mouth" of the inequality symbol opens toward the larger value
Tip 2: Strict inequalities (< >) use open circles, inclusive inequalities (≤ ≥) use closed circles
Tip 3: Less than always shades to the left, greater than always shades to the right
Tip 4: Test values in different regions to verify your solution is correct
Tip 5: Remember that multiplying or dividing by a negative number flips the inequality sign
Common errors: Flipping the inequality symbol incorrectly, confusing open/closed circles, shading in wrong direction, forgetting to include boundary for ≤ or ≥.
Memory aids: "Less than points left", "Greater than points right", "Equal sign means filled dot", "The mouth eats the bigger number".
Essential rules to remember:

• Circle rule: Open circle for < and >, closed circle for ≤ and ≥

• Shading rule: Less than shades left, greater than shades right

• Sign flipping: Multiply/divide by negative number reverses inequality

• Interval notation: Parentheses exclude, brackets include endpoints

• Verification: Always test a solution value to confirm correctness

Visualization: Inequality Solutions
Exercise 6: Multiple Inequalities
Visualizing different inequality types:
\(f_1(x): x < 5\) (shaded left)
\(f_2(x): x \geq 3\) (shaded right, includes 3)
\(f_3(x): x \leq -2\) (shaded left, includes -2)
\(f_4(x): x > -4\) (shaded right)

Analysis: The chart shows how different inequality types create different solution regions.

  • Strict inequalities create open boundaries
  • Inclusive inequalities create closed boundaries
  • Direction of shading depends on the comparison
  • Each inequality represents a half-line on the number line

Questions & Answers

Question: I get confused about when to use open circles versus closed circles. How do I remember which is which?

Answer: Here's how to remember:

Open circle: Used for strict inequalities (\(<\) or \(>\)). The boundary point is NOT included in the solution.

Closed circle: Used for inclusive inequalities (\(\leq\) or \(\geq\)). The boundary point IS included in the solution.

Memory aids:

  • "Less than or equal" has an extra line under the \(<\), so the circle is "filled in" (closed)
  • "Greater than or equal" has an extra line under the \(>\), so the circle is "filled in" (closed)
  • Plain \(<\) or \(>\) have no extra line, so the circle is "empty" (open)

Examples:

  • \(x < 5\): Open circle at 5 (5 is not part of the solution)
  • \(x \leq 5\): Closed circle at 5 (5 is part of the solution)
  • \(x > 3\): Open circle at 3 (3 is not part of the solution)
  • \(x \geq 3\): Closed circle at 3 (3 is part of the solution)

Question: How do I know which direction to shade on the number line?

Answer: The direction of shading depends on the inequality symbol:

Shade to the left: When the variable is less than the boundary value

  • \(x < 5\): Shade left (values smaller than 5)
  • \(x \leq 5\): Shade left (values smaller than or equal to 5)

Shade to the right: When the variable is greater than the boundary value

  • \(x > 5\): Shade right (values larger than 5)
  • \(x \geq 5\): Shade right (values larger than or equal to 5)

Memory aids:

  • "Less than" points to the left (the symbol < looks like an arrow pointing left)
  • "Greater than" points to the right (the symbol > looks like an arrow pointing right)
  • On the number line, smaller numbers are on the left, larger numbers are on the right

Always test a value in your shaded region to verify it satisfies the original inequality!

Question: What's the difference between an equation and an inequality? Why do inequalities have more solutions?

Answer: The key differences are:

Equations:

  • Use equality symbol (=)
  • Usually have one or a finite number of solutions
  • Represent exact values where expressions are equal
  • Example: \(x + 2 = 5\) has one solution: \(x = 3\)

Inequalities:

  • Use inequality symbols (<, >, ≤, ≥)
  • Have infinitely many solutions forming a range or interval
  • Represent a range of values where one expression is larger/smaller than another
  • Example: \(x + 2 > 5\) has infinitely many solutions: all \(x > 3\)

Inequalities represent ranges because they allow for comparisons rather than exact equality. Instead of asking "what value makes this exactly equal?", they ask "what values make this larger/smaller than a given value?"

This is why inequalities are so useful in real-world situations where exact values aren't required, but ranges are important (like minimum requirements, maximum limits, etc.).