1.4 sign charts and inequalities i
TRANSCRIPT
Inequalities
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Sign-Charts and Inequalities IGiven an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
Sign-Charts and Inequalities IGiven an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
Sign-Charts and Inequalities I
For polynomials or rational expressions, factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Sign-Charts and Inequalities I
For polynomials or rational expressions, factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Hence, for x = -3/2:(-3/2 – 3)(-3/2 + 1)
Sign-Charts and Inequalities I
For polynomials or rational expressions, factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Hence, for x = -3/2:(-3/2 – 3)(-3/2 + 1) is (–)(–) = + .
Sign-Charts and Inequalities I
For polynomials or rational expressions, factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Hence, for x = -3/2:(-3/2 – 3)(-3/2 + 1) is (–)(–) = + .
And for x = -1/2:(-1/2 – 3)(-1/2 + 1)
Sign-Charts and Inequalities I
For polynomials or rational expressions, factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Hence, for x = -3/2:(-3/2 – 3)(-3/2 + 1) is (–)(–) = + .
And for x = -1/2:(-1/2 – 3)(-1/2 + 1) is (–)(+) = – .
Sign-Charts and Inequalities I
For polynomials or rational expressions, factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
Sign-Charts and Inequalities IExample B. Determine whether the outcome is
x2 – 2x – 3x2 + x – 2
if x = -3/2, -1/2.+ or – for
Example B. Determine whether the outcome isx2 – 2x – 3x2 + x – 2
x2 – 2x – 3x2 + x – 2
In factored form =(x – 3)(x + 1)(x – 1)(x + 2)
Sign-Charts and Inequalities I
if x = -3/2, -1/2.+ or – for
x2 – 2x – 3x2 + x – 2
In factored form =(x – 3)(x + 1)(x – 1)(x + 2)
Hence, for x = -3/2:(x – 3)(x + 1)(x – 1)(x + 2)
=(–)(–)(–)(+)
Sign-Charts and Inequalities IExample B. Determine whether the outcome is
x2 – 2x – 3x2 + x – 2
if x = -3/2, -1/2.+ or – for
x2 – 2x – 3x2 + x – 2
In factored form =(x – 3)(x + 1)(x – 1)(x + 2)
Hence, for x = -3/2:(x – 3)(x + 1)(x – 1)(x + 2)
=(–)(–)(–)(+)
< 0
Sign-Charts and Inequalities IExample B. Determine whether the outcome is
x2 – 2x – 3x2 + x – 2
if x = -3/2, -1/2.+ or – for
x2 – 2x – 3x2 + x – 2
In factored form =(x – 3)(x + 1)(x – 1)(x + 2)
Hence, 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)
=(–)(+)(–)(+)
Sign-Charts and Inequalities IExample B. Determine whether the outcome is
x2 – 2x – 3x2 + x – 2
if x = -3/2, -1/2.+ or – for
x2 – 2x – 3x2 + x – 2
In factored form =(x – 3)(x + 1)(x – 1)(x + 2)
Hence, 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
Sign-Charts and Inequalities IExample B. Determine whether the outcome is
x2 – 2x – 3x2 + x – 2
if x = -3/2, -1/2.+ or – for
x2 – 2x – 3x2 + x – 2
In factored form =(x – 3)(x + 1)(x – 1)(x + 2)
Hence, 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 IExample B. Determine whether the outcome is
x2 – 2x – 3x2 + x – 2
if x = -3/2, -1/2.+ or – for
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +– – – – x – 1
Sign-Charts and Inequalities I
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +– – – – x – 1
The "+" indicates the region where the output is positive i.e. if 1 < x.
Sign-Charts and Inequalities I
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +– – – – x – 1
The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1.
Sign-Charts and Inequalities I
Construction of the sign-chart of f.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +– – – – x – 1
The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1.
Sign-Charts and Inequalities I
Construction of the sign-chart of f.I. Solve for f = 0 (and denominator = 0) if there is any denominator.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +– – – – x – 1
The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1.
Sign-Charts and Inequalities I
Construction of the sign-chart of f.I. Solve for f = 0 (and denominator = 0) if there is any denominator.II. Draw the real line, mark off the answers from I.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +– – – – x – 1
The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1.
Sign-Charts and Inequalities I
Construction of the sign-chart of f.I. Solve for f = 0 (and denominator = 0) if there is any denominator.II. Draw the real line, mark off the answers from I.III. Sample each segment for signs by testing a point in each segment.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +– – – – x – 1
The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1.
Sign-Charts and Inequalities I
Construction of the sign-chart of f.I. Solve for f = 0 (and denominator = 0) if there is any denominator.II. Draw the real line, mark off the answers from I.III. Sample each segment for signs by testing a point in each segment.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +– – – – x – 1
The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1.
Fact: The sign stays the same for x's in between the values from step I (where f = 0 or f is undefined.)
Sign-Charts and Inequalities I
Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0.
Sign-Charts and Inequalities I
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
Sign-Charts and Inequalities I
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 , -1Mark off these points on a line:
(x-4)(x+1) 4-1
Sign-Charts and Inequalities I
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 , -1Mark off these points on a line:
(x-4)(x+1)
Select points to sample in each segment:4-1
Sign-Charts and Inequalities I
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 , -1Mark off these points on a line:
(x-4)(x+1) 4-1
Select points to sample in each segment:
Test x = - 2,
-2
Sign-Charts and Inequalities I
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 , -1Mark off these points on a line:
(x-4)(x+1) 4-1
Select points to sample in each segment:
Test x = - 2,
get – * – = + .Hence the segment is positive. Draw +sign over it.
-2
Sign-Charts and Inequalities I
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 , -1Mark off these points on a line:
(x-4)(x+1) + + + + + 4-1
Select points to sample in each segment:
Test x = - 2,
get – * – = + .Hence the segment is positive. Draw +sign over it.
-2
Sign-Charts and Inequalities I
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 , -1Mark 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.
Sign-Charts and Inequalities I
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 , -1Mark 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.
Sign-Charts and Inequalities I
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 , -1Mark 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 segmentis positive.Put + over it.
5
Sign-Charts and Inequalities I
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 , -1Mark 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 segmentis positive.Put + over it.
5
Sign-Charts and Inequalities I
Example D. Make the sign chart of f =(x – 3)
(x – 1)(x + 2)
Sign-Charts and Inequalities I
Example D. Make the sign chart of f =(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator which is x = 3.
Sign-Charts and Inequalities I
Example D. Make the sign chart of f =(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).
Sign-Charts and Inequalities I
Example D. Make the sign chart of f =(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.
Sign-Charts and Inequalities I
Example D. Make the sign chart of f =(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
Sign-Charts and Inequalities I
Example D. Make the sign chart of f =
Select a point to sample in each segment:
(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 0 2 4
Sign-Charts and Inequalities I
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
Sign-Charts and Inequalities I
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.
Sign-Charts and Inequalities I
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.
= –
Sign-Charts and Inequalities I
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
The easiest way to solve a polynomial or rational inequality is to use the sign-chart.
Sign-Charts and Inequalities I
Example E. Solve x2 – 3x > 4
The easiest way to solve a polynomial or rational inequality is to use the sign-chart.
Sign-Charts and Inequalities I
Example E. 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,
Sign-Charts and Inequalities I
Example E. 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
Sign-Charts and Inequalities I
Example E. 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
Sign-Charts and Inequalities I
Example E. 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.
Sign-Charts and Inequalities I
Example E. 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.
Sign-Charts and Inequalities I
Example E. 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.
Sign-Charts and Inequalities I
Example E. Solve x2 – 3x > 4
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,
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
Example E. Solve x2 – 3x > 4
0 4-1-2 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,
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
Example E. Solve x2 – 3x > 4
0 4-1-2 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,
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
Example E. Solve x2 – 3x > 4
0 4-1-2 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,
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
Example E. Solve x2 – 3x > 4
0 4-1-2 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,
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
Example E. Solve x2 – 3x > 4
0 4-1-2 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,III. read off the answer from 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
Example E. Solve x2 – 3x > 4
0 4-1
The solutions are the + regions:-2 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,III. read off the answer from 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
Example E. Solve x2 – 3x > 4
0 4-1
The solutions are the + regions: (–∞, –1) U (4, ∞)-2 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.
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
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. 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.
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
Example F. Solve x – 22 < x – 1
3Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (x – 2)(x – 1)
2(x – 1) – 3(x – 2)
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (x – 2)(x – 1)
2(x – 1) – 3(x – 2) = (x – 2)(x – 1)– x + 4
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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, and it’s undefined at x = 1, 2.
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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.
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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.
41 2
UDF UDF
(x – 2)(x – 1)– x + 4
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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.
410 523/2 3
UDF UDF
(x – 2)(x – 1)– x + 4
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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
Sign-Charts and Inequalities I
Example F. Solve x – 22 < x – 1
3
Set the inequality to 0, x – 22
x – 13 < 0
Put the expression into factored form,
x – 22
x – 13 = (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
Inequalities