1.4 “solving linear inequalities”
DESCRIPTION
1.4 “solving linear inequalities”. ex 1:. Graph x > 3. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10. ex 1:. Graph x > 3. - PowerPoint PPT PresentationTRANSCRIPT
Graph x > 3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
ex 1:
Graph x > 3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
ex 1:
Graph x > 3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
ex 1:
“All real numbers greater than three”
Graph x –2 ex 2:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Graph x –2 ex 2:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Graph x –2
“All real numbers less than or equal to negative
two”
ex 2:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve 5x – 8 < 12ex 3:
Solve 5x – 8 < 12 +8 +8
ex 3:
Solve 5x – 8 < 12 +8 +85x < 20
ex 3:
Solve 5x – 8 < 12 +8 +85x < 20 5 5
ex 3:
Solve 5x – 8 < 12 +8 +85x < 20 5 5
ex 3:
x < 4
Solve 5x – 8 < 12 +8 +85x < 20 5 5
ex 3:
x < 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve 5x – 8 < 12 +8 +85x < 20 5 5
ex 3:
x < 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve –3x 12 ex 4:
Solve –3x 12 –3 –3
ex 4:
Solve –3x 12 –3 –3
ex 4:
x –4
Solve –3x 12 –3 –3
When you multiply or divide by a negative ,
the inequality sign switches direction.
ex 4:
x –4
Solve 2x + 1 6x – 23ex 5:
Solve 2x + 1 6x – 23
–2x –2x
ex 5:
Solve 2x + 1 6x – 23
–2x –2x 1 4x – 23
ex 5:
Solve 2x + 1 6x – 23
–2x –2x 1 4x – 23
+23 +23
ex 5:
Solve 2x + 1 6x – 23
–2x –2x 1 4x – 23
+23 +23 24 4x
ex 5:
Solve 2x + 1 6x – 23
–2x –2x 1 4x – 23
+23 +23 24 4x 4 4
ex 5:
Solve 2x + 1 6x – 23
–2x –2x 1 4x – 23
+23 +23 24 4x 4 4
6 x
ex 5:
Solve 2x + 1 6x – 23
–2x –2x 1 4x – 23
+23 +23 24 4x 4 4
6 x
ex 5:
VARIABLES ON THE LEFT-SIDE
!
Solve 2x + 1 6x – 23
–2x –2x 1 4x – 23
+23 +23 24 4x 4 4
6 x
ex 5:
VARIABLES ON THE LEFT-SIDE
!
x 6
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Graph x > –5
ex 6:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Graph x > –5
ex 6:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Graph x > –5
Graph x < 7
ex 6:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Graph x > –5
Graph x < 7
ex 6:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Graph x > –5
Graph x < 7
Graph x > –5 and x < 7
ex 6:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Graph x > –5
Graph x < 7
Graph x > –5 and x < 7
ex 6:
Two simple inequalities joined by
“and” or “or”
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Two simple inequalities joined by
“and” or “or” x > –5 and x < 7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Two simple inequalities joined by
“and” or “or” x > –5 and x < 7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Two simple inequalities joined by
“and” or “or” x > –5 and x < 7
“All real #s that are greater than negative five and less than seven”
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Two simple inequalities joined by
“and” or “or” x > –5 and x < 7
“All real #s that are greater than negative five and less than seven”
–5 < x < 7
Two simple inequalities joined by
“and” or “or”
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Two simple inequalities joined by
“and” or “or” x < –5 or x >7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Two simple inequalities joined by
“and” or “or” x < –5 or x >7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Two simple inequalities joined by
“and” or “or” x < –5 or x >7
“All real #s that are less than negative five or greater than seven”
Solve –2 3x – 8 4ex 7:
Solve –2 3x – 8 4 +8 +8 +8
ex 7:
Solve –2 3x – 8 4 +8 +8 +8
6 3x 12
ex 7:
Solve –2 3x – 8 4 +8 +8 +8
6 3x 12 3 3 3
ex 7:
Solve –2 3x – 8 4 +8 +8 +8
6 3x 12 3 3 3
2 x 4
ex 7:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve –2 3x – 8 4 +8 +8 +8
6 3x 12 3 3 3
x 2 and x 4
2 x 4
ex 7:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve –2 3x – 8 4 +8 +8 +8
6 3x 12 3 3 3 x 2
and x 4 2 x 4
ex 7:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve –2 3x – 8 4 +8 +8 +8
6 3x 12 3 3 3
“All real #s that are greater than or equal to two and less than or equal to four”
x 2 and x 4 2 x 4
ex 7:
Solve 2x + 3 < 1 or 4x – 7 > 5
ex 8:
Solve 2x + 3 < 1 or 4x – 7 > 5 2x + 3 < 1 or 4x – 7 > 5
ex 8:
Solve 2x + 3 < 1 or 4x – 7 > 5 2x + 3 < 1 or 4x – 7 > 5
–3 –3 +7 +7
ex 8:
Solve 2x + 3 < 1 or 4x – 7 > 5 2x + 3 < 1 or 4x – 7 > 5
–3 –3 +7 +7
2x < –2 or 4x > 12
ex 8:
Solve 2x + 3 < 1 or 4x – 7 > 5 2x + 3 < 1 or 4x – 7 > 5
–3 –3 +7 +7
2x < –2 or 4x > 12
2 2 4 4
ex 8:
Solve 2x + 3 < 1 or 4x – 7 > 5 2x + 3 < 1 or 4x – 7 > 5
–3 –3 +7 +7
2x < –2 or 4x > 12
2 2 4 4
ex 8:
x < –1 or x > 3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve 2x + 3 < 1 or 4x – 7 > 5 2x + 3 < 1 or 4x – 7 > 5
–3 –3 +7 +7
2x < –2 or 4x > 12
2 2 4 4
ex 8:
x < –1 or x > 3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve 2x + 3 < 1 or 4x – 7 > 5 2x + 3 < 1 or 4x – 7 > 5
–3 –3 +7 +7
2x < –2 or 4x > 12
2 2 4 4
ex 8:
x < –1 or x > 3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve 2x + 3 < 1 or 4x – 7 > 5 2x + 3 < 1 or 4x – 7 > 5
–3 –3 +7 +7
2x < –2 or 4x > 12
2 2 4 4“All real #s that are less than negative
one or greater than three”
ex 8:
x < –1 or x > 3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
1] Graph –2 x 5
2] Is x = 9 a solution of – x – 2 –4 ?
3] Solve 5 – 2x 27 4] Solve x + 5 < 2
SOLVE & GRAPH :5] 3x + 2 –10 or 2x – 4 –4
6] –19 3x – 4 2
13
92
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
1] Graph –2 x 5
2] Is c = 9 a solution of – x – 2 –4 ?
3] Solve 5 – 2x 27 4] Solve x + 5 < 2
SOLVE & GRAPH :5] 3x + 2 –10 or 2x – 4 –4
6] –19 3x – 4 2
YES ; –5 –4
x –11
2x<
3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x < –4 or x > 0
x –5 and x 2
13
92