Download Solving Inequalities and Interval Notation Workshop

Tutoring Center 972 860 2974
Room: C206
INEQUALITIES AND INTERVAL NOTATION
Solving Inequalities and Interval Notation Workshop
I.
Notation
a. Line graph
b. Set-builder (a, b), [a, b]
c. Interval (x│a ≤ x ≤ b) translated as “x defined as…”
II.
Inequalities
a. Less than
b. Greater than
c. Less than or equals to
d. Greater than or equals to
III.
Solving inequalities
a. Less than/ greater than
b. Less then or equals to/ greater than or equals to
c. Absolute value (solve separately and linearly)
d. Absolute valueless than 0 (= no sol.)
e. More practice examples
Line graph:
Graph these on a number line.
1. x > 7
(7, ∞) or {x│x>7}
2. x < 2
(-∞, 2) or {x│x<2}
3. x ≥ -4
[4, ∞) or {x│x ≥-4}
4. x ≤ 3
(-∞, 3] or {x│x≤3}
5. x ≥ 4 and x ≤ 10
[4, 10] or {x│4≤x≤10}
6. x ≥ -2 and x < 5
[-2, 5) or {x│-2≤x<5}
7. x ≥ 8 and x ≤ 4
(-∞, 4] U [8, ∞) or
{x│x ≤4}or{x│x ≥8}
8. x > 7 and x ≤ 0
(-∞, 0] U (7, ∞) or
{x│x ≤0}or{x│x >7}
Solving Inequalities – just like solving equations except for –multiplication and -division
x+3>8
-3 -3
x>5
x – 7 < 12
+7 +7
x < 19
2x + 3 ≥ 7
-3 -3
2x ≥ 4
x≥2
3x – 9 ≤ 27
+9 +9
3x ≤ 36
x ≤ 12
x/3 + 2 ≤ 8
- 2 -2
x/3 ≤ 6
x ≤ 18
When multiplying or dividing by a negative number, flip the inequality.
-2x + 3 ≥ 7
-3x – 9 ≤ 27
-3 -3
+9 +9
-2x ≥ 4
-3x ≤ 36
x ≤ -2
x ≥ -12
What about absolute value?
│x + 4│> 8
x + 4 > 8 or x + 4 < -8
-4 -4
- 4 -4
x >4
x < -12
x/-3 + 2 ≤ 8
- 2 -2
x/-3 ≤ 6
x ≥ -3
│x│+ 4 > 8
-4 -4
│x│> 4
x > 4 or x < 4
-4 + x /-2 ≥ 10
+4
+4
x /-2 ≥ 14
x ≤ -28
│3x - 2│≥ 0
3x – 2 ≥ 0 or
+ 2 +2
3x ≥ 2
x ≥ 2/3
│-2x + 5│≤ 0 has no solution
Others to try to solve, graph, and put in set-builder notation and interval notation:
1. - 4x - 8 ≥ 12
x ≤ -5
2. │x + 4│- 2 > 8
x > 6, x < -14
3. 2x/-5 + 2 ≤ 12
x ≥ -25
4. -5x + 5 ≤ 30
x ≥ -5
5. 6 +│4x - 1│≥ 9
x ≥ 1, x ≤ -1 / 2
6. -5 + x /-3 ≥ -18
x ≤ 36
7. -3x > 21
x < -7
8. 5x + 2 ≥ 3x – 8
x ≥ -5
9. 4x/-3 + 7 ≤ 9x + 4
x ≥ 9 / 31
10. 2(x -3) + 5x < 8(x -1)
x>2
11. 2(x +5) ≤ 5x – 3x +14
10 ≤ 14, x = all real numbers
12. 3(x +5) – 2x ≤ x + 12
15 ≤ 12, x has no solutions
3x – 2 ≤ 0
+ 2 +2
3x ≤ 2
x ≤ 2/3