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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:
4264 x
1. Solve the following inequality and sketch
the interval that satisfies it:
2210 xSubtract 6 from each component of the equation…
4264 x
1. Solve the following inequality and sketch
the interval that satisfies it:
2210 xSubtract 6 from each component of the equation…
Divide each component by -2 …remembering to switch the signs.
15 x
4264 x
1. Solve the following inequality and sketch
the interval that satisfies it:
2210 x
51 x
Subtract 6 from each component of the equation…
Divide each component by -2 …remembering to switch the signs.
15 xOr
4264 x
1. Solve the following inequality and sketch
the interval that satisfies it:
2210 x
51 x
Subtract 6 from each component of the equation…
Divide each component by -2 …remembering to switch the signs.
15 xOr
4264 x
Sketch the interval:
2-4. A conjunctive inequality is the intersection of two intervals.
2. Solve and sketch the interval.
x264
3. Solve and sketch the interval.426 x
2-4. A conjunctive inequality is the intersection of two intervals.
x210
2. Solve and sketch the interval.
x264 Subtract 6
3. Solve and sketch the interval.426 x
2-4. A conjunctive inequality is the intersection of two intervals.
x210
x5
2. Solve and sketch the interval.
x264 Subtract 6
Divide by -2
3. Solve and sketch the interval.426 x
2-4. A conjunctive inequality is the intersection of two intervals.
x210
5xx5
2. Solve and sketch the interval.
x264 Subtract 6
Divide by -2
3. Solve and sketch the interval.426 x
Or
2-4. A conjunctive inequality is the intersection of two intervals.
x210
5xx5
2. Solve and sketch the interval.
x264 Subtract 6
Divide by -2
3. Solve and sketch the interval.426 x
Or
2-4. A conjunctive inequality is the intersection of two intervals.
x210
5xx5
2. Solve and sketch the interval.
x264 Subtract 6
Divide by -2
3. Solve and sketch the interval.426 x
Subtract 6 22 xDivide by -2 1x
Or
2-3. Solve and sketch each of the two inequalities separately.
x210
5xx5
2. Solve and sketch the interval.
x264 Subtract 6
Divide by -2
3. Solve and sketch the interval.426 x
Subtract 6 22 xDivide by -2 1x
Or
4. The solution to the compound inequality is the intersection of each separate inequality.
426 x
x264
Each inequality separately:
4. The solution to the compound inequality is the intersection of each separate inequality.
426 x
x264
x264
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.
AND
426 x
Each inequality separately:
4. The solution to the compound inequality is the intersection of each separate inequality.
426 x
x264
x264
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.
AND
426 x
Each inequality separately:
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
63 x
01x
0163 xORx
1. Solve and sketch the interval.
2. Solve and sketch the interval.
Solve
63 x
01x
2x
0163 xORx
1. Solve and sketch the interval.
2. Solve and sketch the interval.
Divide by 3
Solve
63 x
01x
2x
0163 xORx
1. Solve and sketch the interval.
2. Solve and sketch the interval.
Divide by 3
Solve
63 x
01x
2x
0163 xORx
1. Solve and sketch the interval.
2. Solve and sketch the interval.
1x
Divide by 3
Add 1 …
Solve
63 x
01x
2x
0163 xORx
1. Solve and sketch the interval.
2. Solve and sketch the interval.
1x
Divide by 3
Add 1 …
Solve 0163 xORx
The ‘or’ means that either the left inequality needs to be true OR the right inequality needs to be true.
Solve 0163 xORx
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 0163 xORx
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 0163 xORx
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, ∞).
