2. The easiest way to solve a polynomial or rational
inequality is to use the sign–chart.
Inequalities and Sign Charts
3. Example A. Solve x2 – 3x > 4
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart.
Inequalities and Sign Charts
4. Example A. Solve x2 – 3x > 4
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
Inequalities and Sign Charts
5. Example A. Solve x2 – 3x > 4
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
Setting one side to 0, we have x2 – 3x – 4 > 0
Inequalities and Sign Charts
6. Example A. Solve x2 – 3x > 4
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
II. factor the expression
Setting one side to 0, we have x2 – 3x – 4 > 0
Inequalities and Sign Charts
7. Example A. Solve x2 – 3x > 4
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
II. factor the expression
Setting one side to 0, we have x2 – 3x – 4 > 0 or
(x – 4)(x + 1) > 0.
Inequalities and Sign Charts
8. Example A. Solve x2 – 3x > 4
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
II. factor the expression and draw the sign–chart,
Setting one side to 0, we have x2 – 3x – 4 > 0 or
(x – 4)(x + 1) > 0.
Inequalities and Sign Charts
9. Example A. Solve x2 – 3x > 4
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
II. factor the expression and draw the sign–chart,
Setting one side to 0, we have x2 – 3x – 4 > 0 or
(x – 4)(x + 1) > 0. The roots are x = –1, 4 both are
odd power roots.
Inequalities and Sign Charts
10. Example A. Solve x2 – 3x > 4
4 (odd power)
–1 (odd power) 5
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
II. factor the expression and draw the sign–chart,
(x – 4)(x + 1)
+ + + – – – – – – + + + +
Setting one side to 0, we have x2 – 3x – 4 > 0 or
(x – 4)(x + 1) > 0. The roots are x = –1, 4 both are
odd power roots. Sample x = 5, we get +,
completing the signs:
Inequalities and Sign Charts
11. Example A. Solve x2 – 3x > 4
4 (odd power)
–1 (odd power) 5
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
II. factor the expression and draw the sign–chart,
(x – 4)(x + 1)
+ + + – – – – – – + + + +
Setting one side to 0, we have x2 – 3x – 4 > 0 or
(x – 4)(x + 1) > 0. The roots are x = –1, 4 both are
odd power roots. Sample x = 5, we get +,
completing the signs:
Inequalities and Sign Charts
12. Example A. Solve x2 – 3x > 4
4 (odd power)
–1 (odd power)
The solutions are the + regions: (–∞, –1) U (4, ∞)
5
4
–1
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart. To do this,
I. set one side of the inequality to 0,
II. factor the expression and draw the sign–chart,
III. read off the answer from the sign chart.
(x – 4)(x + 1)
+ + + – – – – – – + + + +
Setting one side to 0, we have x2 – 3x – 4 > 0 or
(x – 4)(x + 1) > 0. The roots are x = –1, 4 both are
odd power roots. Sample x = 5, we get +,
completing the signs:
Inequalities and Sign Charts
13. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Inequalities and Sign Charts
14. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
15. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression:
16. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
17. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
18. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered).
19. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered).
20. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ".
x=0 (even) x=2 (odd)
x= 1 (even) x=3
21. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even) x=3
22. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even)
+ change
x=3
23. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even)
+ change
unchanged
+
x=3
24. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even)
+ change
unchanged
unchanged
+
+
x=3
25. Inequalities and Sign Charts
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example B. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even)
+ change
unchanged
unchanged
+
+
Hence the solution is 2 < x.
x=3
27. Example C. Solve x – 2
2 <
x – 1
3
Set the inequality to 0, x – 2
2
x – 1
3
< 0
Inequalities and Sign Charts
28. Example C. Solve x – 2
2 <
x – 1
3
Set the inequality to 0, x – 2
2
x – 1
3
< 0
Put the expression into factored form,
Inequalities and Sign Charts
29. Example C. Solve x – 2
2 <
x – 1
3
Set the inequality to 0, x – 2
2
x – 1
3
< 0
Put the expression into factored form,
x – 2
2
x – 1
3
=
(x – 2)(x – 1)
2(x – 1) – 3(x – 2)
Inequalities and Sign Charts
30. Example C. Solve x – 2
2 <
x – 1
3
Set the inequality to 0, x – 2
2
x – 1
3
< 0
Put the expression into factored form,
x – 2
2
x – 1
3
=
(x – 2)(x – 1)
2(x – 1) – 3(x – 2)
=
(x – 2)(x – 1)
– x + 4
Inequalities and Sign Charts
31. Example C. Solve x – 2
2 <
x – 1
3
Set the inequality to 0, x – 2
2
x – 1
3
< 0
Put the expression into factored form,
x – 2
2
x – 1
3
=
(x – 2)(x – 1)
2(x – 1) – 3(x – 2)
=
(x – 2)(x – 1)
– x + 4
Hence the inequality is (x – 2)(x – 1)
– x + 4 < 0
Inequalities and Sign Charts
32. Example C. Solve x – 2
2 <
x – 1
3
Set the inequality to 0, x – 2
2
x – 1
3
< 0
Put the expression into factored form,
x – 2
2
x – 1
3
=
(x – 2)(x – 1)
2(x – 1) – 3(x – 2)
=
(x – 2)(x – 1)
– x + 4
Hence the inequality is (x – 2)(x – 1)
– x + 4 < 0
It has a root at x = 4, it's undefined at x = 1, 2
and all of them are of odd–power.
4
1 2
UDF UDF
(x – 2)(x – 1)
– x + 4
Inequalities and Sign Charts
33. Example C. Solve x – 2
2 <
x – 1
3
Set the inequality to 0, x – 2
2
x – 1
3
< 0
Put the expression into factored form,
x – 2
2
x – 1
3
=
(x – 2)(x – 1)
2(x – 1) – 3(x – 2)
=
(x – 2)(x – 1)
– x + 4
Hence the inequality is (x – 2)(x – 1)
– x + 4 < 0
It has a root at x = 4, it's undefined at x = 1, 2
and all of them are of odd–power.
Testing x = 5, we get –, so the sign–chart is:
4
1
0 5
+ + + – – + + + + – – – –
2
UDF UDF
(x – 2)(x – 1)
– x + 4
Inequalities and Sign Charts
34. Example C. Solve x – 2
2 <
x – 1
3
Set the inequality to 0, x – 2
2
x – 1
3
< 0
Put the expression into factored form,
x – 2
2
x – 1
3
=
(x – 2)(x – 1)
2(x – 1) – 3(x – 2)
=
(x – 2)(x – 1)
– x + 4
Hence the inequality is (x – 2)(x – 1)
– x + 4 < 0
It has a root at x = 4, it's undefined at x = 1, 2
and all of them are of odd–power.
Testing x = 5, we get –, so the sign–chart is:
4
1
0 5
+ + + – – + + + + – – – –
2
UDF UDF
(x – 2)(x – 1)
– x + 4
The answer are the shaded intervals or (1, 2) U [4 ∞).
Inequalities and Sign Charts