Solve Inequalities, Find Equations Of Lines, And Analyze Lin

Solve inequalities, find equations of lines, and analyze linear functions

Analyze and solve various algebraic problems including inequalities, equations of lines, slope calculations, and applications such as modeling and predictions. Show all work clearly, justify answers with sentences where needed, and graph solutions accurately.

Paper For Above instruction

Introduction

Linear equations and inequalities serve as foundational concepts in algebra, providing tools for modeling real-world situations and understanding relationships between variables. This paper explores solving inequalities, determining equations of lines, analyzing slopes, and applying these skills to practical problems such as predicting growth trends and calculating interest. Emphasis will be placed on clear problem-solving steps, accurate calculations, and proper graphing techniques.

Solving Linear Inequalities and Compound Inequalities

Consider the inequality: x - 7 > 0.

Solving for x, we add 7 to both sides: x > 7. The solution set in interval notation is (7, ∞), and graphically, the solution is represented as an open circle at 7 with shading to the right.

Next, examine the compound inequality: -20 ≤ 6x - 5

To solve, split into two parts:

  • 6x - 5 ≥ -20 ⇒ 6x ≥ -15 ⇒ x ≥ -\frac{5}{2}
  • 6x - 5

Combined, the solution is: -2.5 ≤ x

Finding Equation of Lines through Points

Given points (−3, 5) and (6, -2):

  1. Calculate the slope:
  2. slope m = (y₂ - y₁) / (x₂ - x₁) = (-2 - 5) / (6 - (−3)) = -7 / 9
  3. Write the equation in point-slope form:
  4. Using point (−3, 5): y - 5 = (−7/9)(x + 3)
  5. Convert to slope-intercept form:
  6. y = (−7/9)x - (7/3) + 5 = (−7/9)x + (8/3)
  7. Standard form (Ax + By = C):
  8. Multiply through by 9: 9y = -7x + 24
  9. Rearranged: 7x + 9y = 24

Graph the line using these points or the slope-intercept form. The slope of the line is −7/9, and the y-intercept is (0, 8/3).

Parallel and Perpendicular Lines

Find the line parallel to y = 2x + 1 passing through (5, 4):

Since parallel lines have equal slopes, the equation is: y - 4 = 2(x - 5) ⇒ y = 2x - 10 + 4 ⇒ y = 2x - 6.

The slope is 2, and the equation is y = 2x - 6.

Drawing a line through (-4, 3) parallel to the x-axis yields a horizontal line:

Equation: y = 3. The slope is 0.

Analyzing Linear Equations and Their Intercepts

Given line: 3x + 4y = 12

  • Slope: rearranged to y = -\(\frac{3}{4}\)x + 3
  • Y-intercept: (0, 3)

Using Point and Slope to Write Equations and Graph

Points: (-3, 5) and (6, -2), slope: −7/9, equation in point-slope form, slope-intercept, and standard forms are found above. Graph these to visualize the line.

Applications and Real-life Modeling

Model the growth of Starbucks stores: in 2005, 8896 stores; in 2014, 11965 stores. Let y be the number of stores, and x be years since 2005 (x=0 at 2005).

The slope m is calculated as:

m = (11965 - 8896) / (9 - 0) = 2070/1 = 2070 stores per year.

Linear model: y = 2070x + 8896.

Predict x=15 (year 2020): y = 2070(15) + 8896 = 31050 + 8896 = 39946 stores.

The slope indicates an average annual increase of approximately 2070 stores, reflecting growth trends in retail expansion.

Interest and Salary Calculations

Simple interest on a $22,000 loan at 9.35% over 6 years:

I = P r t = 22000 0.0935 6 = $12,321.

Andrew’s extension salary after a 3.7% raise:

Raise amount: 72400 * 0.037 = $2,678.80.

New salary: 72400 + 2678.80 = $75,078.80.

Perimeter of a Garden

For a rectangular garden with width w and length l, perimeter P = 2(w + l). If the dimensions are given, substitute and compute perimeter.

Solving Inequalities with Graphing

Solve inequalities such as x - 3 > 0, which simplifies to x > 3, with solution in interval notation (3, ∞). For inequalities involving fractions, multiply through by the LCD to clear fractions and solve accordingly. Graph the solutions with proper notation.

Graphing Linear Equations

Using a table of values, find four (x, y) pairs satisfying the equations. For example, for 28x + 7y = 15:

  • Choose x values, solve for y, and record pairs.

Plot the points and draw the line accordingly, ensuring all work is shown to validate solutions.

Conclusion

This comprehensive approach enhances understanding of algebraic concepts, supports problem-solving proficiency, and illustrates the application of mathematics in real-world contexts. Accurate calculations, detailed steps, and clear explanations are essential for mastery.

References

  • Algebra and Trigonometry. Robert F. Blitzer. Pearson, 2020.
  • Elementary Algebra. Harold R. Jacobs. W. H. Freeman, 2018.
  • Linear Algebra and Its Applications. Gilbert Strang. Cengage Learning, 2019.
  • Precalculus: Mathematics for Calculus. Stewart, Redlin, Watson. Cengage, 2018.
  • Applied Mathematics for the Managerial, Life, and Social Sciences. Soo T. Tan. Pearson, 2015.
  • College Algebra. Margaret L. Lial, John Hornsby. Pearson, 2018.
  • Mathematics for Business and Economics. Charles David Miller. McGraw-Hill, 2019.
  • Finite Mathematics and Applied Calculus. Sheldon Ross. Pearson, 2021.
  • Basic College Mathematics. Mark Dugopolski. Pearson, 2019.
  • Introductory Business Statistics. Prem S. Mann. Wiley, 2019.

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