1. x + 2 > 9
x + 2 - 2 > 9 - 2
x > 7
2. x - 4 < 10
x - 4 + 4 < 10 + 4
x < 14
3. -5x < 25
(-5x) / -5 > 25 / -5
x > -5
4. -3x + 7 ≥ 31
-3x + 7 - 7 ≥ 31 - 7
-3x ≥ 24
(-3x) / -3 ≤ 24 / -3
x ≤ -8
5. 8 > 4x
8 / 4 > (4x) / 4
2 > x
6. 10 - 4 < -2x
6 < -2x
6 / -2 > (-2x) / -2
-3 > x
1. x > 7 (Open circle at 7, shading to the right)
2. x < 14 (Open circle at 14, shading to the left)
3. x > -5 (Open circle at -5, shading to the right)
4. x ≤ -8 (Closed circle at -8, shading to the left)
5. x < 2 (Open circle at 2, shading to the left)
6. x < -3 (Open circle at -3, shading to the left)
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Answers & Comments
Answer:
Let's solve each inequality step by step:
1. x + 2 > 9
Subtract 2 from both sides:
x + 2 - 2 > 9 - 2
x > 7
The solution set is x > 7. On a number line, we represent this as an open circle at 7 and shading to the right.
2. x - 4 < 10
Add 4 to both sides:
x - 4 + 4 < 10 + 4
x < 14
The solution set is x < 14. On a number line, we represent this as an open circle at 14 and shading to the left.
3. -5x < 25
Divide both sides by -5, remembering to flip the inequality symbol when dividing by a negative number:
(-5x) / -5 > 25 / -5
x > -5
The solution set is x > -5. On a number line, we represent this as an open circle at -5 and shading to the right.
4. -3x + 7 ≥ 31
Subtract 7 from both sides:
-3x + 7 - 7 ≥ 31 - 7
-3x ≥ 24
Divide both sides by -3, remembering to flip the inequality symbol:
(-3x) / -3 ≤ 24 / -3
x ≤ -8
The solution set is x ≤ -8. On a number line, we represent this as a closed circle at -8 and shading to the left.
5. 8 > 4x
Divide both sides by 4:
8 / 4 > (4x) / 4
2 > x
The solution set is x < 2. On a number line, we represent this as an open circle at 2 and shading to the left.
6. 10 - 4 < -2x
Simplify the left side:
6 < -2x
Divide both sides by -2, remembering to flip the inequality symbol:
6 / -2 > (-2x) / -2
-3 > x
The solution set is x < -3. On a number line, we represent this as an open circle at -3 and shading to the left.
Here is a summary of the solution sets and their corresponding number line representations:
1. x > 7 (Open circle at 7, shading to the right)
2. x < 14 (Open circle at 14, shading to the left)
3. x > -5 (Open circle at -5, shading to the right)
4. x ≤ -8 (Closed circle at -8, shading to the left)
5. x < 2 (Open circle at 2, shading to the left)
6. x < -3 (Open circle at -3, shading to the left)