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Chapter P: Prerequisite Information. Section P-7: Solving Inequalities Algebraically and Graphically. Objectives. You will learn about: Solving absolute value inequalities Solving quadratic inequalities Approximating solutions to inequalities Projectile motion Why:

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Chapter P: Prerequisite Information

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Chapter p prerequisite information

Chapter P:Prerequisite Information

Section P-7:

Solving Inequalities Algebraically and Graphically


Objectives

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


Vocabulary

Vocabulary

  • Union of two sets A and B

  • Projectile motion


Solving absolute value inequalities

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


Example 1 solve an absolute value inequality

Example 1:Solve an Absolute Value Inequality

  • Solve |x – 4|< 8


Example 2 solving another absolute value inequality

Example 2:Solving another absolute value inequality

  • Solve |3x – 2|≥ 5


Example 3 solving a quadratic inequality

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.


Example 4 solving another quadratic inequality

Example 4:Solving another quadratic inequality

  • Solve 2x2 + 3x ≤ 20

  • Again, solve the equation and graph


Example 5 solving another quadratic inequality

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.


Example 6 showing there is no solution

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.


Example 7 solving a cubic inequality

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.


Projectile motion

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


Example 8 finding the height of a projectile

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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