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Mastering Inequalities: Solving Graphically and Algebraically

This chapter focuses on solving inequalities both algebraically and graphically. It covers crucial techniques, including solving absolute value and quadratic inequalities, approximating solutions, and exploring projectile motion. You will learn how to represent inequalities graphically and analyze real-world applications. Through various examples, we’ll clarify the solutions to absolute value inequalities, quadratic functions, and even cubic inequalities, providing you with the essential tools to tackle complex problems.

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Mastering Inequalities: Solving Graphically and Algebraically

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  1. Chapter P:Prerequisite Information Section P-7: Solving Inequalities Algebraically and Graphically

  2. Objectives • You will learn about: • Solving absolute value inequalities • Solving quadratic inequalities • Approximating solutions to inequalities • Projectile motion • Why: • These techniques are involved in using a graphing utility to solve inequalities

  3. Vocabulary • Union of two sets A and B • Projectile motion

  4. Solving Absolute Value Inequalities • Let u be an algebraic expression in x and let a be a real number with a ≥ 0. • If |u| < a, then u is in the interval (-a, a). • That is, |u| < a if and only if –a < u < a • If |u| > a, then u is in the interval (-∞, -a) or (a, ∞). • That is |u| > a if and only if u < -a or u > a

  5. Example 1:Solve an Absolute Value Inequality • Solve |x – 4|< 8

  6. Example 2:Solving another absolute value inequality • Solve |3x – 2|≥ 5

  7. Example 3:Solving a Quadratic Inequality • Solve x2 – x – 12 > 0 • First, solve the equation x2 – x – 12 = 0 • Graph the equation and observe where the graph is above zero.

  8. Example 4:Solving another quadratic inequality • Solve 2x2 + 3x ≤ 20 • Again, solve the equation and graph

  9. Example 5:Solving another quadratic inequality • Solve x2– 4x + 1 > 0 • This one we must do graphically because the equation does not factor. • Enter on your calculator: y = x2 – 4x + 1 • We will use the trace key to observe where we have zeros.

  10. Example 6:Showing there is no solution • Solve x2+ 2x + 2 ≤ 0. • Graph the equation. Where is the equation below the x-axis? • Use the quadratic formula to verify your answer.

  11. Example 7:Solving a cubic inequality • Solve x3 +2x2 + 2 ≥ 0 • Enter the equation in y = • Estimate where your zeros are and then use the zero trace function to find the values.

  12. Projectile Motion • Suppose an object is launched vertically from a point s0 feet above the ground with an initial velocity v0 feet per second. The vertical position s (in feet) of the object t seconds after it is launched is: • s = -16t2 + v0t + s0

  13. Example 8:Finding the height of a projectile • A projectile is launched straight up from ground level with an initial velocity of 288 ft/sec. • When will the projectile’s height above the ground be1152 ft? • When will the projectile’s height above the ground be at least 1152 ft?

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