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Questions and Answers
Which ordered pair is a valid solution to the inequality $3x + 2y \leq 12$?
For the inequality $4x - y \geq 5$, which ordered pair satisfies this condition?
What inequality represents the maximum price of the headphones, $y$, if the cost of the tablet, $x$, does not exceed $900$?
Which statement correctly describes the graph of the inequality $y < 8x - 10$?
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Which ordered pair does not satisfy the inequality $-2x + 3y < 6$?
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Which of the following points is a solution to the linear inequality $5x + 4y < 120$?
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Study Notes
###Â Linear Inequalities: Standard Form
- Coach Taylor has a budget of 120forsportsdrinksandenergybars,whereeachenergybarcosts120 for sports drinks and energy bars, where each energy bar costs 120forsportsdrinksandenergybars,whereeachenergybarcosts5 and each sports drink costs $4.
- The linear inequality in standard form for this situation is: 5x + 4y ≤ 120, where x is the number of energy bars and y is the number of sports drinks.
- If Coach Taylor buys no energy bars (x = 0), she can buy a maximum of 30 sports drinks (y = 30).
- If Coach Taylor buys no sports drinks (y = 0), she can buy a maximum of 24 energy bars (x = 24).
- Coach Taylor would not be able to afford 10 energy bars and 10 sports drinks because the total cost would be 90,exceedingherbudgetof90, exceeding her budget of 90,exceedingherbudgetof120.
- The graph of the inequality 5x + 4y ≤ 120 is a shaded region below a solid line, representing all possible combinations of energy bars and sports drinks that are within Coach Taylor's budget.
Linear Inequality: Slope-Intercept Form
- The inequality representing the graph is: y ≤ -5/4x + 10.
- To determine the inequality, identify the slope and y-intercept of the line.
- The line is solid and the shaded region is below the line, indicating that the inequality sign is ≤.
Linear Inequality: Application to Solar Energy
- Alyssa generates 240 watts per solar panel (x) but loses 12 watts per minute of battery charging (y).
- She wants her net energy (240x - 12y) to be no more than 12,000 watts.
- If she uses 40 solar panels and spends 300 minutes charging, she will have a net energy of 9,600 watts, which is less than her limit of 12,000 watts.
- The inequality in slope-intercept form is y ≥ 20x - 1,000.
Solving Linear Inequalities
- To check if a point is a solution to a given linear inequality, substitute the values of the point into the inequality.
- If the inequality is true after substitution, the point is a solution.
- For example, the point (-5, -5) is not a solution to the inequality 2x + 3 > 1, because substituting the values results in -10 + 3 > 1, which is false.
Writing Linear Inequalities from Context
- Samantha spent no more than $900 on a tablet (x) and headphones (y).
- The inequality representing this situation is x + y ≤ 900.
Graphing Linear Inequalities
- When graphing a linear inequality, the type of line (solid or dashed) and the shaded region depend on the inequality sign.
- A solid line represents ≤ or ≥, while a dashed line represents < or >.
- The shaded region is above the line for inequalities with > or ≥, and below the line for inequalities with < or ≤.
- For example, the inequality 7y > x - 9 has a dashed line and the shaded region above the line.
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Description
This quiz explores the concept of linear inequalities using a real-life scenario involving budgeting for sports drinks and energy bars. You'll learn how to derive the inequality, understand the graphical representation, and practice determining feasible combinations within a given budget.
