sign charts and inequalities

1. x2 – 2x – 3
x2 + x – 2
In factored form =
(x – 3)(x + 1)
(x – 1)(x + 2)
So for x = –3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(–)
(–)(+)
< 0
For x = –1/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(+)
(–)(+)
> 0
This leads to the sign charts of formulas. The sign–
chart of a formula gives the signs of the outputs.
Sign–Charts and Inequalities I
Example B. Determine whether the outcome is
x2 – 2x – 3
x2 + x – 2
if x = –3/2, –1/2.+ or – for
For polynomials or rational expressions,
factor them to determine the signs of their outputs.

2. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0, and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1) + + + + + – – – – – + + + + +
0 4–1
Select points to sample in each segment:
Test x = – 2,
get – * – = + .
Hence the segment
is positive. Draw +
sign over it.
–2
Test x = 0,
get – * + = –.
Hence this segment
is negative.
Put – over it.
Test x = 5,
get + * + = +.
Hence this segment
is positive.
Put + over it.
5
Sign–Charts and Inequalities I

3. Example D. Make the sign chart of f =
Select a point to sample in each segment:
Test x = –3,
we've a
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Mark these values on a real line.
(x – 3)
(x – 1)(x + 2) –2 1 3
UDF UDF f=0
–3
( – )
( – )( – )
= –
segment.
0 2 4
Test x = 0,
we've a
( – )
( – )( + )
= +
segment.
Test x = 2,
we've a
( – )
( + )( + )
segment.
= –
Test x = 4,
we've a
( + )
( + )( + )
segment.
= +
– – – – + + + – – – + + + +
Sign–Charts and Inequalities I

4. Example E. Solve x2 – 3x > 4
0 4–1
The solutions are the + regions: (–∞, –1) U (4, ∞)
–2 5
4–1
Note: The empty dot means those numbers are excluded.
The easiest way to solve a polynomial or rational
inequality is to use the sign–chart.
Draw the sign–chart, sample the points x = –2, 0, 5
(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.
Sign–Charts and Inequalities I

5. Example F. 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
Draw the sign chart by sampling x = 0, 3/2, 3, 5
It has a root at x = 4, and it's undefined at x = 1, 2.
410 5
+ + + – – + + + + – – – –
23/2 3
UDF UDF
(x – 2)(x – 1)
– x + 4
The answer are the shaded negative regions,
i.e. (1, 2) U [4 ∞).
Sign–Charts and Inequalities I

6. Sign-Charts and Inequalities
Exercise A. Draw the sign–charts of the following
formulas.
1. (x – 2)(x + 3)
4. (2 – x)(x + 3) 5. –x(x + 3)
7. (x + 3)2
9. x(2x – 1)(3 – x)
12. x2(2x – 1)2(3 – x)
13. x2(2x – 1)2(3 – x)2 14. x2 – 2x – 3
16. 1 –15. x4 – 2x3 – 3x2
(x – 2)
(x + 3)2.
(2 – x)
(x + 3)3.
–x
(x + 3)6.
8. –4(x + 3)4
x
(3 – x)(2x – 1)10.
11. x2(2x – 1)(3 – x)
1
x + 3
17. 2 – 2
x – 2
18. 1
2x + 1 19. –1
x + 3
– 1 2
x – 2
20. –2
x – 4
1
x + 2

7. Sign-Charts and Inequalities
Exercise B. Use the sign–charts method to solve the
following inequalities.
1. (x – 2)(x + 3) > 0
3. (2 – x)(x + 3) ≥ 0
8. x2(2x – 1)2(3 – x) ≤ 0
9. x2 – 2x < 3
14. 1 <13. x4 > 4x2
(2 – x)
(x + 3)2.
–x
(x + 3)4.
7. x2(2x – 1)(3 – x) ≥ 0
1
x 15. 2 2
x – 2
16. 1
x + 3
2
x – 2
17. >2
x – 4
1
x + 2
5. x(x – 2)(x + 3)
x
(x – 2)(x + 3)6. ≥ 0
10. x2 + 2x > 8
11. x3 – 2x2 < 3x 12. 2x3 < x2 + 6x
≥
≥ 0
≤ 0
≤ 0
≤ 18. 1 < 1
x2

8. C. Solve the inequalities, use the answers from Ex.1.3.
Inequalities

10. (Answers to odd problems) Exercise A.
1.
3.
Sign-Charts and Inequalities
x = 2x = –3
+ – +
x = 2x = –3
– + –UDF
5.
x = 0x = –3
– + –
7.
x = –3
+ +
9.
x = 1/2x = 0 x = 3
+ – + –

11. Sign-Charts and Inequalities
11.
x = 1/2x = 0 x = 3
– – + –
13.
x = 1/2x = 0 x = 3
+ + + +
15.
x = 0x = -1 x = 3
+ – – +
17.
x = 3x = 2
+ – +UDF
19.
x = -3
+ – + –
x = -8 x = 2
UDF UDF

12. Sign-Charts and Inequalities
Exercise B.
1. (–∞, 3) ∪ (2, ∞) 3. [–3, 2]
5. (–∞, 3] ∪ [0, 2]
9. (–1, 3)7. {0} ∪ [1/2, 3] 11. (–∞, 1) ∪ (0, 3)
13. (–∞, –2) ∪ (2, ∞) 15. (–∞, –2) ∪ (2, ∞)
17. (–8, –2) ∪ (4, ∞)
Exercise C.
1. The statement is not true
3. (–∞, – 12/5) ∪ (2, ∞)
5. {12/5} ∪ [1, ∞)