3.6 Solve Applications with Linear Inequalities

Linear inequalities help us solve real-world problems with constraints. We'll learn how to translate scenarios into mathematical language, using symbols like ≤ and ≥ to represent "at least" or "no more than." This skill is crucial for modeling situations with upper or lower limits.

Once we've written our inequality, we'll solve it algebraically and graphically. We'll interpret the solutions in context, making sure they make sense for the original problem. This process bridges math and practical applications, showing how inequalities apply to everyday situations.

Translating and Solving Linear Inequality Applications

Translation of real-world scenarios

Top images from around the web for Translation of real-world scenarios

• 

  Applications of Linear Functions | Boundless Algebra View original

  Is this image relevant?

• 

  Linear Inequalities and Systems of Linear Inequalities in Two Variables | College Algebra: Co ... View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Linear Inequalities (one variable) View original

  Is this image relevant?

• 

  Applications of Linear Functions | Boundless Algebra View original

  Is this image relevant?

• 

  Linear Inequalities and Systems of Linear Inequalities in Two Variables | College Algebra: Co ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Translation of real-world scenarios

• 

  Applications of Linear Functions | Boundless Algebra View original

  Is this image relevant?

• 

  Linear Inequalities and Systems of Linear Inequalities in Two Variables | College Algebra: Co ... View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Linear Inequalities (one variable) View original

  Is this image relevant?

• 

  Applications of Linear Functions | Boundless Algebra View original

  Is this image relevant?

• 

  Linear Inequalities and Systems of Linear Inequalities in Two Variables | College Algebra: Co ... View original

  Is this image relevant?

1 of 3

• Identify the unknown quantity and represent it with a variable (x, y, or z)
• Determine the constraints or conditions in the scenario
  • Look for key phrases indicating inequalities ("at least," "at most," "no more than," or "no less than")
  • Convert these phrases into mathematical symbols ($\leq$, $\geq$, $[<](https://www.fiveableKeyTerm:<)$, or $[>](https://www.fiveableKeyTerm:>)$)
• Write an inequality representing the situation
  • Express the constraints using the variable and the appropriate inequality symbol
  • Combine multiple constraints, if necessary, using "and" or "or" conditions (minimum age and maximum weight)
• Recognize when a real-world application requires a compound inequality to accurately represent the situation

Solutions for practical inequalities

• Isolate the variable on one side of the inequality
  • Add or subtract the same value from both sides to eliminate constants
  • Multiply or divide both sides by the same positive value to eliminate coefficients
  • If multiplying or dividing by a negative value, reverse the inequality symbol
• Solve the resulting inequality
  • For $ax + b < c$ or $ax + b > c$, solve for $x$ to get the solution in the form $x < \frac{c-b}{a}$ or $x > \frac{c-b}{a}$
  • For $ax + b \leq c$ or $ax + b \geq c$, solve for $x$ to get the solution in the form $x \leq \frac{c-b}{a}$ or $x \geq \frac{c-b}{a}$
• Graph the solution on a number line (graphical representation)
  • Use an open circle for strict inequalities ($<$ or $>$)
  • Use a closed circle for inclusive inequalities ($\leq$ or $\geq$)
  • Shade the region that satisfies the inequality (values greater than 5, less than or equal to 10)
• Express the solution set using interval notation

Interpretation of inequality solutions

• Relate the solution back to the original problem
  • Describe the solution using the context of the problem (minimum age requirement, maximum weight limit)
  • Identify the range of values that satisfy the conditions given in the scenario (between 18 and 65 years old)
• Determine if the solution makes sense in the context of the problem
  • Check if the solution adheres to the constraints of the scenario (within budget, meets safety standards)
  • Verify that the units of the solution are appropriate for the context (hours, dollars, pounds)
• Communicate the solution clearly and concisely
  • Use appropriate units and terminology from the original problem (gallons of gas, number of tickets sold)
  • Provide a written explanation of the solution that addresses the question or goal of the scenario

Additional Concepts in Linear Inequalities

• Understanding the role of variables in representing unknown quantities in inequalities
• Recognizing the solution set as all possible values that satisfy the inequality
• Using algebraic manipulation techniques to solve complex inequalities
• Applying linear inequalities to model and solve real-world problems in various fields