1.6 Solving Compound Inequalities

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```					1.6 Solving Compound
Inequalities
Understanding that conjunctive
inequalities take intersections of
intervals and disjunctive inequalities
take unions of intervals
Part 1: First Box of the Worksheet

Part 1 of the presentation will go over
completing the first box of the worksheet:

“Understanding conjunctive compound
inequalities…”
1. Solve the following inequality and sketch
the interval that satisfies it:

 4  6  2x  4
1. Solve the following inequality and sketch
the interval that satisfies it:

 4  6  2x  4
Subtract 6 from each component of the equation…

10  2x  2
1. Solve the following inequality and sketch
the interval that satisfies it:

 4  6  2x  4
Subtract 6 from each component of the equation…

10  2x  2
Divide each component by -2 …remembering to switch the signs.

5  x 1
1. Solve the following inequality and sketch
the interval that satisfies it:

 4  6  2x  4
Subtract 6 from each component of the equation…

10  2x  2
Divide each component by -2 …remembering to switch the signs.

5  x 1
Or   1 x  5
1. Solve the following inequality and sketch
the interval that satisfies it:

 4  6  2x  4
Subtract 6 from each component of the equation…

10  2x  2
Divide each component by -2 …remembering to switch the signs.

5  x 1
Or   1 x  5
Sketch the interval:
2-4. A conjunctive inequality is the
intersection of two intervals.

2. Solve and sketch the interval.
 4  6  2x

3. Solve and sketch the interval.
6  2x  4
2-4. A conjunctive inequality is the
intersection of two intervals.

2. Solve and sketch the interval.
 4  6  2x
Subtract 6
10  2x

3. Solve and sketch the interval.
6  2x  4
2-4. A conjunctive inequality is the
intersection of two intervals.

2. Solve and sketch the interval.
 4  6  2x
Subtract 6
10  2x
Divide by -2
5 x

3. Solve and sketch the interval.
6  2x  4
2-4. A conjunctive inequality is the
intersection of two intervals.

2. Solve and sketch the interval.
 4  6  2x
Subtract 6
10  2x
Divide by -2
5 x
Or
x5
3. Solve and sketch the interval.
6  2x  4
2-4. A conjunctive inequality is the
intersection of two intervals.

2. Solve and sketch the interval.
 4  6  2x
Subtract 6
10  2x
Divide by -2
5 x
Or
x5
3. Solve and sketch the interval.
6  2x  4
2-4. A conjunctive inequality is the
intersection of two intervals.

2. Solve and sketch the interval.
 4  6  2x
Subtract 6
10  2x
Divide by -2
5 x
Or
x5
3. Solve and sketch the interval.
6  2x  4
Subtract 6         2x  2
Divide by -2        x 1
2-3. Solve and sketch each of the two
inequalities separately.

2. Solve and sketch the interval.
 4  6  2x
Subtract 6
10  2x
Divide by -2
5 x
Or
x5
3. Solve and sketch the interval.
6  2x  4
Subtract 6         2x  2
Divide by -2        x 1
4. The solution to the compound inequality is the
intersection of each separate inequality.

Each inequality separately:

 4  6  2x

6  2x  4
4. The solution to the compound inequality is the
intersection of each separate inequality.

Each inequality separately:

 4  6  2x

6  2x  4
Taking the intersection means considering points that satisfy
BOTH inequalities. The first inequality needs to be true and
the second one needs to be true.
 4  6  2x
AND
6  2x  4
4. The solution to the compound inequality is the
intersection of each separate inequality.

Each inequality separately:

 4  6  2x

6  2x  4
Taking the intersection means considering points that satisfy
BOTH inequalities. The first inequality needs to be true and
the second one needs to be true.
 4  6  2x
AND
6  2x  4
Part 1: First Box of the Worksheet

The point of this exercise is that you can solve the
conjunctive compound inequality in two
different ways:
1. Solving algebraically by performing operations
on all 3 parts of the compound inequality.
2. Solve each simple inequality separately and
then taking the intersection of the two intervals.
Part 1: First Box of the Worksheet

The point of this exercise is that you can solve the
conjunctive compound inequality in two
different ways:
1. Solving algebraically by performing operations
on all 3 parts of the compound inequality.
2. Solve each simple inequality separately and
then taking the intersection of the two intervals.

The first way is easier for solving conjunctive
inequalities. However, disjunctive inequalities can
only be solved by solving each simple inequality
separately and then taking their union.
Part 2: Second Box of the Worksheet

Part 2 of the presentation will go over using
unions to solve disjunctive compound
inequalities:

“Solving disjunctive compound inequalities…”
Solve           3x  6 OR x  1  0

1. Solve and sketch the interval.

3x  6

2. Solve and sketch the interval.

x 1 0
Solve             3x  6 OR x  1  0

1. Solve and sketch the interval.

3x  6
Divide by 3
x  2

2. Solve and sketch the interval.

x 1 0
Solve             3x  6 OR x  1  0

1. Solve and sketch the interval.

3x  6
Divide by 3
x  2

2. Solve and sketch the interval.

x 1 0
Solve             3x  6 OR x  1  0

1. Solve and sketch the interval.

3x  6
Divide by 3
x  2

2. Solve and sketch the interval.

x 1 0
x 1
Solve             3x  6 OR x  1  0

1. Solve and sketch the interval.

3x  6
Divide by 3
x  2

2. Solve and sketch the interval.

x 1 0
x 1
Solve          3x  6 OR x  1  0

The ‘or’ means that either the left inequality needs to
be true OR the right inequality needs to be true.
Solve          3x  6 OR x  1  0

The ‘or’ means that either the left inequality needs to
be true OR the right inequality needs to be true.

Thus, we want all the points that make the left
inequality true together with all the points that make
the right inequality true – we take the union of the two
intervals.

The union is:
Solve          3x  6 OR x  1  0

The ‘or’ means that either the left inequality needs to
be true OR the right inequality needs to be true.

Thus, we want all the points that make the left
inequality true together with all the points that make
the right inequality true – we take the union of the two
intervals.

The union is:
Solve           3x  6 OR x  1  0

The ‘or’ means that either the left inequality needs to
be true OR the right inequality needs to be true.

Thus, we want all the points that make the left
inequality true together with all the points that make
the right inequality true – we take the union of the two
intervals.

The union is:
In interval notation this is: (-∞,2](1, ∞).

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