2.2.5 Representing equations, inequality on lines, graphs

• What is the general purpose of inequality symbols in mathematics?

  To compare numbers or expressions
• In a real-world example, an open dot at 140 cm indicates that someone taller than 140 cm can ride a roller coaster.


  True
• The symbol "≤" allows for the possibility that two values could be equal.


  True
• If the inequality includes the symbol < or ≤, the line is extended to the left.


  True
• If x ≥ 5, the number 5 is included in the solution.

  True
• In the inequality x ≥ 7, which direction is the line extended on the number line?

  Right
• What does the symbol ">" mean in inequalities?

  Greater than
• A closed dot on a number line means the boundary point is included in the solution.


  True
• If x ≤ 7, the number 7 is included in the solution.


  True
• Match the type of dot with its correct meaning:

  Open dot ↔️ Boundary point not included

  Closed dot ↔️ Boundary point included
• Order the steps for representing the solution of an inequality on a number line.

  1️⃣ Plot the boundary point

  2️⃣ Determine the extension direction based on the inequality type

  3️⃣ Extend the line to the appropriate direction
• An open dot on a number line indicates that the boundary point is included in the solution.

  False
• How do you represent x ≥ 7 on a number line?

  Closed dot at 7
• In the inequality "x < 5," what does the symbol "<" tell us about x?

  x must be smaller than 5
• What is the title of the study material?

  Inequality Regions
• What symbols are used to indicate whether a boundary point is included in the solution of an inequality?

  Parentheses and brackets
• What notation is used to represent x > 3?

  (3, ∞)
• The symbol "<" is used to indicate that one value is strictly less than another.


  True
• What does the symbol '∞' represent in interval notation?

  Infinity
• The interval notation for x > 3 is (3, ∞)


  True
• What is the equation for the line parallel to the x-axis?

  $y =$$x - 1$
• The symbol "≠" indicates that two values cannot be the same.


  True
• What is the purpose of interval notation?

  Express ranges of solutions
• What is the first inequality given in the study material?

  x ≥ -2
• How do you represent x < 3 on a number line?

  Open dot at 3
• What type of line is used for ≤ or ≥ when graphing inequalities?

  Solid
• The shaded region in the graph 'R' satisfies all three inequalities simultaneously.


  True
• What is the value of x?

  $x =$$1$
• How can the inequality regions be represented graphically?

  • The inequalities can be graphed on a coordinate plane
  • The regions satisfying the inequalities are shaded
  • The boundaries of the regions are determined by the equality versions of the inequalities
• How can the inequality regions be used to solve optimization problems?

  • The feasible region defined by the inequality constraints can be identified
  • The objective function (e.g. maximizing profit, minimizing cost) can be optimized within the feasible region
  • The optimal solution corresponds to the point in the feasible region that maximizes/minimizes the objective function
• Dashed lines are used for inequalities with < or > symbols when graphing.


  True
• Which region is shaded when graphing multiple inequalities?

  The region that satisfies all
• What is the equation for the line in the image?

  $x +$$y =$$4$
• What is the third inequality given in the study material?

  4x + 3y < 12
• What does the symbol '[' indicate in interval notation?

  Includes boundary
• What does the symbol '(' indicate in interval notation?

  Excludes boundary
• How can the inequality regions be used to model real-world decision-making problems?

  • The inequalities can represent constraints such as resource limitations, budget constraints, or production capacity
  • The feasible region defined by the inequalities represents the set of possible solutions that satisfy all the constraints
  • The optimal solution within the feasible region corresponds to the best decision that maximizes the desired outcome
• What are the key differences between linear programming and inequality regions?

  • Linear programming involves optimizing a linear objective function subject to linear constraints
  • Inequality regions focus on graphically representing the feasible region defined by the constraints, without necessarily optimizing an objective function
  • Linear programming is a more general technique that can be applied to a wider range of optimization problems
• What does the area marked C in the graph represent?

  Solution region
• If you are taller than 140 cm to ride a roller coaster, you extend the line to the right of 140 on the number line.


  True