Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Solve quadratic inequalities algebraically and graphically.
• Solve inequalities involving rational expressions.
• Formulate and solve word problems using inequalities.
• Use interval notation to represent solutions.
• Sketch and use graphs to solve quadratic inequalities.

Lesson summary

This week, we delve deeper into the world of equations and inequalities, building upon the foundational knowledge acquired in earlier grades and weeks. Equations and inequalities are fundamental tools in mathematics, allowing us to model real-world scenarios, solve problems related to finance, engineering, and everyday life, and make informed decisions. For South African learners, understanding these concepts is crucial for navigating financial planning (budgeting, loans, investments), understanding statistical data presented in the news (economic trends, unemployment rates), and even making informed decisions about resource allocation in their communities.

Lesson notes

2.1 Quadratic Inequalities A quadratic inequality is an inequality that can be written in the form ax 2 + bx + c > 0, ax 2 + bx + c 2 + bx + c ≥ 0, or ax 2 + bx + c ≤ 0, where a, b, and c are real numbers and a ≠

0. Solving Quadratic Inequalities Algebraically: Rewrite the inequality: Manipulate the inequality to have zero on one side. For example, x 2 > 3x + 4 becomes x 2 - 3x - 4 >

0. Find the roots: Solve the corresponding quadratic equation ax 2 + bx + c =

0. You can use factoring, the quadratic formula, or completing the square. These roots are the critical values.

Determine the intervals: The roots divide the number line into intervals.

Test each interval: Choose a test value from each interval and substitute it into the original inequality. If the inequality is true for the test value, the entire interval is part of the solution.

Write the solution: Express the solution in interval notation. Pay attention to whether the endpoints are included (for ≤ or ≥) or excluded (for ). Solving Quadratic Inequalities Graphically: Sketch the graph: Sketch the graph of the quadratic function y = ax 2 + bx + c.

Identify the roots: Find the x-intercepts (roots) of the graph. These are the same roots found algebraically.

Determine the intervals: Identify the intervals where the graph is above the x-axis (for > 0) or below the x-axis (for 2 - 5x + 6 2 - 5x + 6 =

0. This factors to (x - 2)(x - 3) = 0, so x = 2 or x =

3. Intervals: The roots divide the number line into the intervals (-∞, 2), (2, 3), and (3, ∞).

Test Values: Interval (-∞, 2): Test x = 0. (0) 2 - 5(0) + 6 = 6, which is not less than

0. So, this interval is not part of the solution. Interval (2, 3): Test x = 2.5. (2.5) 2 - 5(2.5) + 6 = -0.25, which is less than

0. So, this interval is part of the solution. Interval (3, ∞): Test x = 4. (4) 2 - 5(4) + 6 = 2, which is not less than

0. So, this interval is not part of the solution.

Solution: The solution is (2, 3). Note that the endpoints are not included because the inequality is strictly less than.