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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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inequality that can be put in
one of the forms

where a, b, and c are real numbers and a≠0.

Step 1: write the inequality in standard form.
Step 2: solve the related quadratic equation.
Step 3: locate the boundary points on a
number line.
Step 4: construct a sign chart.
Step 5: the solution set is the interval that
produced a true statement.

Inequality

Solve:

Solution:
Step 1: Write the inequality in
standard form.

Step 2: Solve the related quadratic equation.

(x-1)(x-3)=0

x+1=0 or x-3=0

x=-1 or x= 3

The boundary points are -1 and 3.

These two points divide the number
line into three test intervals, namely
(−∞,-1) ,(−1,3) ,and (3,∞) .

Take a test point within each
interval and check the sign.

 Test Interval Test Point X+1 X-3 (x+1)(x-3) (−∞,-1) -2 - - +> 0 (−1,3) 0 + + -< 0 (3,∞) 4 + + +>0

The question is "where ?"

Our table shows that

in the interval (-1,3).

So the solution set is
the interval (-1,3)

Practice Exercises

Solving Rational Inequalities

Solve:

It is incorrect to multiply both sides by
x-2 to clear fractions. The problem is
that x-2 contains a variable and can be
positive or negative, depending on the
value of x. Thus, we do not know
whether or not to reverse the sense of
inequality.

Example 2:

solution: We begin by finding values
of that make the numerator and
denominator 0

Set the numerator and denominator equal to 0.

Solve

x=-5 and x=2

The boundary points are -5 and 2.

Locate boundary points -5 and 2
on a number line.

These boundary points divide the
number line into three intervals,
namely (−∞,-5),(-5,2), and (2,∞).
Now, construct a sign chart: take
one test point from each interval
and check the signs.

 Intervals Test Points x+5 x-2 (−∞,-5) -6 - - +> 0 (-5,2) 0 + - -< 0 (2,∞) 3 + + +>0

The given question is "for what values
of the quantity is positive?"

Our table shows that for

(−∞,-5) or (2,∞).

Thus the solution set is

(−∞,-5) or (2,∞).

The graph of the solution set on a
number line is shown as follows:

Practice Exercises

 Solve: Answers: (-5,-2) (−∞,3) or [4,∞) [-4,-2]