Linear Inequalities in Standard Form

### 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.