3 3 absolute inequalities-geom-x
Transcript of 3 3 absolute inequalities-geom-x
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method.
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method.
Example A. Translate the meaning of |x| < 7 and draw x.
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line.
Example A. Translate the meaning of |x| < 7 and draw x.
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line. |x| means “the distance between x and 0”,
Example A. Translate the meaning of |x| < 7 and draw x.
Example A. Translate the meaning of |x| < 7 and draw x.
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line. |x| means “the distance between x and 0”, so the expression |x| < c means “the distance between x and 0 is smaller than c” (provided that c is not negative in which case no such x exists).
Example A. Translate the meaning of |x| < 7 and draw x.
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line. |x| means “the distance between x and 0”, so the expression |x| < c means “the distance between x and 0 is smaller than c” (provided that c is not negative in which case no such x exists).
|x| < 7
the distance between x and 0
Example A. Translate the meaning of |x| < 7 and draw x.
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line. |x| means “the distance between x and 0”, so the expression |x| < c means “the distance between x and 0 is smaller than c” (provided that c is not negative in which case no such x exists).
|x| < 7
the distance between x and 0 is less than 7.
Example A. Translate the meaning of |x| < 7 and draw x.
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line.
These are all the numbers which are within 7 units from 0, from –7 to 7.
|x| means “the distance between x and 0”, so the expression |x| < c means “the distance between x and 0 is smaller than c” (provided that c is not negative in which case no such x exists).
|x| < 7
the distance between x and 0 is less than 7.
Example A. Translate the meaning of |x| < 7 and draw x.
–7 < x < 7-7-7 70
x
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line.
These are all the numbers which are within 7 units from 0, from –7 to 7.
x
|x| means “the distance between x and 0”, so the expression |x| < c means “the distance between x and 0 is smaller than c” (provided that c is not negative in which case no such x exists).
|x| < 7
the distance between x and 0 is less than 7.
Example A. Translate the meaning of |x| < 7 and draw x.
–7 < x < 7-7-7 70
x
Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line.
These are all the numbers which are within 7 units from 0, from –7 to 7.
x
The open circles means the end points are not included in the solution.
|x| means “the distance between x and 0”, so the expression |x| < c means “the distance between x and 0 is smaller than c” (provided that c is not negative in which case no such x exists).
|x| < 7
the distance between x and 0 is less than 7.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
Example B. Translate the meaning of |x – 2| < 3 and solve.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
|x – 2| < 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
|x – 2| < 3
the distance between x and 2
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
|x – 2| < 3
the distance between x and 2 less than 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
|x – 2| < 3
the distance between x and 2 less than 3
2
Translate the symbols to a geometric description.
Draw
Example B. Translate the meaning of |x – 2| < 3 and solve.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
|x – 2| < 3
the distance between x and 2 less than 3
2
x xright 3left 3
Translate the symbols to a geometric description.
Draw
Example B. Translate the meaning of |x – 2| < 3 and solve.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
|x – 2| < 3
the distance between x and 2 less than 3
–1 52
x xright 3left 3
Translate the symbols to a geometric description.
Draw
Example B. Translate the meaning of |x – 2| < 3 and solve.
I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.
Absolute Value Inequalities
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.
|x – 2| < 3
the distance between x and 2 less than 3
Hence –1 < x < 5.–1 52
x xright 3left 3
Translate the symbols to a geometric description.
Draw
Example B. Translate the meaning of |x – 2| < 3 and solve.
Absolute Value Inequalities
Example C. Translate the meaning of |x| ≥ 7 and draw.
I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.
Absolute Value Inequalities
The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”.
Example C. Translate the meaning of |x| ≥ 7 and draw.
I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.
Absolute Value Inequalities
The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”. In picture
Example C. Translate the meaning of |x| ≥ 7 and draw.
-7-7 70x x
I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.
end point included
Absolute Value Inequalities
The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”. In picture
Example C. Translate the meaning of |x| ≥ 7 and draw.
x < –7 or 7 < x-7-7 70x x
I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.
end point included
Absolute Value Inequalities
The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”. In picture
Note that the answer uses the word “or” for expressing the choices of two sections–not the word “and”.
Example C. Translate the meaning of |x| ≥ 7 and draw.
x < –7 or 7 < x-7-7 70x x
I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.
end point included
Absolute Value Inequalities
The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”. In picture
Note that the answer uses the word “or” for expressing the choices of two sections–not the word “and”. The word “and” is used when there are multiple conditions to be met.
Example C. Translate the meaning of |x| ≥ 7 and draw.
x < –7 or 7 < x-7-7 70x x
I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.
end point included
Absolute Value Inequalities
The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”. In picture
Note that the answer uses the word “or” for expressing the choices of two sections–not the word “and”. The word “and” is used when there are multiple conditions to be met.
Example C. Translate the meaning of |x| ≥ 7 and draw.
x < –7 or 7 < x-7-7 70x x
I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| > c means the “the distance between x and y is more than c”.
end point included
Absolute Value Inequalities
The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”. In picture
Note that the answer uses the word “or” for expressing the choices of two sections–not the word “and”. The word “and” is used when there are multiple conditions to be met.
Example C. Translate the meaning of |x| ≥ 7 and draw.
x < –7 or 7 < x-7-7 70x x
I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| > c means the “the distance between x and y is more than c”. We recall that we may break up | | for multiplication i.e. |x * y| = |x| * |y|.
end point included
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| =
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)|
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2|
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)|
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| We use this step to help us to extract the geometric information.
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric information.
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)|
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
–3/2
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
–3/2
x xright 3/2left 3/2
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
–3 0–3/2
x xright 3/2left 3/2
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2
Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2| |2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
Hence x < –3 or 0 < x.–3 0–3/2
x xright 3/2left 3/2
We use this step to help us to extract the geometric information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2
I. (One Piece | |–Inequalities)
Absolute Value Inequalities Let’s express intervals as absolute value inequalities.
I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.
Absolute Value Inequalities Let’s express intervals as absolute value inequalities.
I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.
Absolute Value Inequalities
–c 0right cleft c
+c|x| < c
Let’s express intervals as absolute value inequalities.
I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.
Absolute Value Inequalities Let’s express intervals as absolute value inequalities.
–c 0right cleft c
Recall that |x – y| translates into “the distance between x and y”
+c|x| < c
I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.
Absolute Value Inequalities Let’s express intervals as absolute value inequalities.
–c 0right cleft c
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”.
+c|x| < c
I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.
Absolute Value Inequalities Let’s express intervals as absolute value inequalities.
–c 0right cleft c
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to write intervals into inequalities. Specifically, we need to locate the midpoint of the given interval first.
+c|x| < c
I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.
Absolute Value Inequalities Let’s express intervals as absolute value inequalities.
–c 0right cleft c
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to write intervals into inequalities. Specifically, we need to locate the midpoint of the given interval first.
+c|x| < c
The mid-point m between two numbers x and y
In picture: o bm
I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.
Absolute Value Inequalities Let’s express intervals as absolute value inequalities.
–c 0right cleft c
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to write intervals into inequalities. Specifically, we need to locate the midpoint of the given interval first.
+c|x| < c
The mid-point m between two numbers x and y is their average
that is m = . a + b2 In picture:
o b(a+b)/2
m
I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.
Absolute Value Inequalities Let’s express intervals as absolute value inequalities.
–c 0right cleft c
Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to write intervals into inequalities. Specifically, we need to locate the midpoint of the given interval first.
+c|x| < c
The mid-point m between two numbers x and y is their average
that is m = . a + b2
For example, the mid-point of 2 and 4 is (2 + 4)/2 = 3.
In picture: o b
(a+b)/2m
2 4(2+4)/2m = 30
Absolute Value Inequalities To write an interval [a, b] into an absolute value inequality,
a b
Example E. Express [2, 4] as an absolute value inequality in x.2 40
Absolute Value Inequalities To write an interval [a, b] into an absolute value inequality, view [a, b] as a “circle”
a b
Example E. Express [2, 4] as an absolute value inequality in x.2 40
Absolute Value Inequalities To write an interval [a, b] into an absolute value inequality, view [a, b] as a “circle” with center at its midpoint m = (a + b)/2,
a bm = (b + a )/2
Example E. Express [2, 4] as an absolute value inequality in x.2 40
Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality, view [a, b] as a “circle” with center at its midpoint m = (a + b)/2, with radius r = (b – a)/2 which is half of the distance from a to b.
2 40
a b
r = (b – a) /2
m = (b + a )/2
Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality, view [a, b] as a “circle” with center at its midpoint m = (a + b)/2, with radius r = (b – a)/2 which is half of the distance from a to b.
2 40
a b
r = (b – a) /2
m = (b + a )/2l x – m l ≤ r
An abs-value inequality
Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality, view [a, b] as a “circle” with center at its midpoint m = (a + b)/2, with radius r = (b – a)/2 which is half of the distance from a to b.
2 4m=30
a b
r = (b – a) /2
m = (b + a )/2l x – m l ≤ r
Viewing [2, 4] as a circle, its center is at m = (2 + 4) / 2 = 3,
An abs-value inequality
Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality, view [a, b] as a “circle” with center at its midpoint m = (a + b)/2, with radius r = (b – a)/2 which is half of the distance from a to b.
2 4m=30
a b
r = (b – a) /2
m = (b + a )/2l x – m l ≤ r
Viewing [2, 4] as a circle, its center is at m = (2 + 4) / 2 = 3,with radius |4 – 2| / 2 = 1
An abs-value inequality
Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality, view [a, b] as a “circle” with center at its midpoint m = (a + b)/2, with radius r = (b – a)/2 which is half of the distance from a to b.
2 4m=30
a b
r = (b – a) /2
m = (b + a )/2l x – m l ≤ r
Viewing [2, 4] as a circle, its center is at m = (2 + 4) / 2 = 3,with radius |4 – 2| / 2 = 1 so the interval is lx – 3l ≤ 1.
An abs-value inequality
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
–c 0right cleft c
+c
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
–c 0right cleft c
+cc < |x|
An abs-value inequality
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
–c 0right cleft c
+cc < |x|
An abs-value inequality
Likewise the expression c < |x – y| means the “the distance between x and y is more than c”.
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
–c 0right cleft c
+cc < |x|
An abs-value inequality
Example F. Express the following two intervals as an absolute value inequality in x.
2 40
Likewise the expression c < |x – y| means the “the distance between x and y is more than c”.
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
–c 0right cleft c
+cc < |x|
An abs-value inequality
Example F. Express the following two intervals as an absolute value inequality in x.
2 40
Let’s find the abs-value inequality of the gap (2, 4).
Likewise the expression c < |x – y| means the “the distance between x and y is more than c”.
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
–c 0right cleft c
+cc < |x|
An abs-value inequality
Example F. Express the following two intervals as an absolute value inequality in x.
2 4m=30
Let’s find the abs-value inequality of the gap (2, 4).The interval (2, 4) has its center at m = (2 + 4) / 2 = 3,with radius (4 – 2)/ 2 = 1
Likewise the expression c < |x – y| means the “the distance between x and y is more than c”.
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
–c 0right cleft c
+cc < |x|
An abs-value inequality
Example F. Express the following two intervals as an absolute value inequality in x.
2 4m=30
Let’s find the abs-value inequality of the gap (2, 4).The interval (2, 4) has its center at m = (2 + 4) / 2 = 3,with radius (4 – 2)/ 2 = 1 so (2, 4) is lx – 3l < 1.
Likewise the expression c < |x – y| means the “the distance between x and y is more than c”.
lx – 3l < 1
Il. (Two–Piece | |–Inequalities)
If x < –c or c < x then c < |x| (c > 0), i.e.
Absolute Value Inequalities
–c 0right cleft c
+cc < |x|
An abs-value inequality
Example F. Express the following two intervals as an absolute value inequality in x.
2 4m=30
Let’s find the abs-value inequality of the gap (2, 4).The interval (2, 4) has its center at m = (2 + 4) / 2 = 3,with radius (4 – 2)/ 2 = 1 so (2, 4) is lx – 3l < 1. So the intervals to the two sides is lx – 3l >1.
Likewise the expression c < |x – y| means the “the distance between x and y is more than c”.
lx – 3l < 1
Ex. A. Translate and solve the expressions geometrically.Draw the solution.1. |x| < 2 2. |x| < 5 3. |–x| < 2 4. |–x| ≤ 5
5. |x| ≥ –2 6. |–2x| < 6 7. |–3x| ≥ 6 8. |–x| ≥ –5 9. |3 – x| ≥ –5 10. |3 + x| ≤ 7 11. |x – 9| < 5
12. |5 – x| < 5 13. |4 + x| ≥ 9 14. |2x + 1| ≥ 3
21. |5 – 2x| ≤ 3 22. |3 + 2x| < 7 23. |–2x + 3| > 5
24. |4 – 2x| ≤ –3 25. |3x + 3| < 5 26. |3x + 1| ≤ 7
Absolute Value Inequalities
15. |x – 2| < 1 16. |3 – x| ≤ 5 17. |x – 5| < 5
18. |7 – x| < 3 19. |8 + x| < 9 20. |x + 1| < 3
Ex. B. Divide by the coefficient of the x–term, translate and solve the expressions geometrically. Draw the solution.
27. |8 – 4x| ≤ 3 28. |4x – 5| < 9 29. |4x + 1| ≤ 9