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# Chapter P: Prerequisite Information - PowerPoint PPT Presentation

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

Section P-7:

Solving Inequalities Algebraically and Graphically

### Objectives

• Solving absolute value inequalities

• Approximating solutions to inequalities

• Projectile motion

• Why:

• These techniques are involved in using a graphing utility to solve inequalities

### Vocabulary

• Union of two sets A and B

• Projectile motion

### 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

• Solve |x – 4|< 8

### Example 2:Solving another absolute value inequality

• Solve |3x – 2|≥ 5

### 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

• Solve 2x2 + 3x ≤ 20

• Again, solve the equation and graph

### 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

• Solve x2+ 2x + 2 ≤ 0.

• Graph the equation. Where is the equation below the x-axis?

### 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

• 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

• 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?