College Algebra

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These numbers divide the real line into the intervals (–∞, 0), (0, 1), (1, 3), (3, ∞)

Slide 52

E.g. 5—Solving an Inequality with Repeated Factors

We make the following diagram, using test points to determine the sign of each factor in each interval.

Slide 53

E.g. 5—Solving an Inequality with Repeated Factors

From the diagram, we see that x(x – 1)2(x – 3) < 0

for x in the interval (0, 1) or (1, 3).

So the solution set is the union of these two intervals: (0, 1)U(1, 3)

Slide 54

E.g. 5—Solving an Inequality with Repeated Factors

The solution is illustrated here.

Slide 55

E.g. 6—An Inequality Involving a Quotient

Solve:

First, we move all nonzero terms to the left side.

Then, we simplify using a common denominator.

Slide 56

E.g. 6—An Inequality Involving a Quotient

Slide 57

E.g. 6—An Inequality Involving a Quotient

The factors of the left-hand side are 2x and 1 – x.

These are zero when x is 0 and 1.

These numbers divide the real line into the intervals (–∞, 0), (0, 1), (1, ∞)

Slide 58

E.g. 6—An Inequality Involving a Quotient

We make the following diagram using test points to determine the sign of each factor in each interval.

We see that the solution set is: [0, 1)

Slide 59

E.g. 6—An Inequality Involving a Quotient

We include the endpoint 0 as the original inequality requires the quotient to be greater than or equal to 1.

However, we do not include the endpoint 1, since the quotient in the inequality is not defined at 1.

Always check the endpoints of solution intervals to determine whether they satisfy the original inequality.

Slide 60

E.g. 6—An Inequality Involving a Quotient

The solution set [0, 1) is illustrated here.

Slide 61

Modeling with Inequalities

Slide 62

Modeling with Inequalities

Modeling real-life problems frequently leads to inequalities.

We are often interested in determining when one quantity is more (or less) than another.