solve each one
x + 3 < 4(x + 1) + 2
2x - 8 ≥ 7 - 3x
3(x + 2) < 14 - x
4(x - 1) < x + 5
5 - 2( x + 1) > x
6(x - 4) ≤ 2x
x + 3 < 4(x + 1) + 2
x + 3 < 4x + 4 + 2
x + 3 < 4x + 6
3 < 3x
1 < x
2x - 8 ≥ 7 - 3x
2x + 3x ≥ 7 + 8
5x ≥ 15
x ≥ 3
3(x + 2) < 14 - x
3x + 6 < 14 - x
4x < 8
x < 2
4(x - 1) < x + 5
4x - 4 < x + 5
3x < 9
x < 3
5 - 2(x + 1) > x
5 - 2x - 2 > x
3 > 3x
1 > x
6(x - 4) ≤ 2x
6x - 24 ≤ 2x
4x ≤ 24
x ≤ 6
1. x + 3 < 4(x + 1) + 2:
Step 1: Distribute the 4 on the right side of the inequality: x + 3 < 4x + 4 + 2
Step 2: Combine like terms: x + 3 < 4x + 6
Step 3: Subtract x from both sides: 3 < 3x + 6
Step 4: Subtract 6 from both sides: -3 < 3x
Step 5: Divide both sides by 3 (remembering to reverse the inequality when dividing by a negative number): -1 < x
2. 2x - 8 ≥ 7 - 3x:
Step 1: Combine like terms: 2x + 3x ≥ 7 + 8
Step 2: Simplify: 5x ≥ 15
Step 3: Divide both sides by 5: x ≥ 3
3. 3(x + 2) < 14 - x:
Step 1: Distribute the 3 on the left side of the inequality: 3x + 6 < 14 - x
Step 2: Add x to both sides: 3x + 6 + x < 14
Step 3: Combine like terms: 4x + 6 < 14
Step 4: Subtract 6 from both sides: 4x < 8
Step 5: Divide both sides by 4: x < 2
4. 4(x - 1) < x + 5:
Step 1: Distribute the 4 on the left side of the inequality: 4x - 4 < x + 5
Step 2: Subtract x from both sides: 4x - x - 4 < 5
Step 3: Combine like terms: 3x - 4 < 5
Step 4: Add 4 to both sides: 3x < 9
Step 5: Divide both sides by 3: x < 3
5. 5 - 2(x + 1) > x:
Step 1: Distribute the -2 on the left side of the inequality: 5 - 2x - 2 > x
Step 2: Combine like terms: 3 - 2x > x
Step 3: Add 2x to both sides: 3 > 3x
Step 4: Divide both sides by 3: 1 > x (or x < 1)
6. 6(x - 4) ≤ 2x:
Step 1: Distribute the 6 on the left side of the inequality: 6x - 24 ≤ 2x
Step 2: Subtract 2x from both sides: 6x - 2x - 24 ≤ 0
Step 3: Combine like terms: 4x - 24 ≤ 0
Step 4: Add 24 to both sides: 4x ≤ 24
Step 5: Divide both sides by 4: x ≤ 6
To solve each of these equations, we will simplify and rearrange the expressions to isolate the variable on one side of the inequality sign. Here's how we can solve each one step by step:
1) x + 3 < 4(x + 1) + 2
First, distribute the 4 on the right side of the equation:
x + 3 < 4x + 4 + 2
Simplify:
x + 3 < 4x + 6
Next, subtract x from both sides to isolate the variable:
3 < 4x - x + 6
Combine like terms:
3 < 3x + 6
Subtract 6 from both sides:
-3 < 3x
Lastly, divide both sides by 3 (since we want to isolate x):
-1 < x
The solution to this equation is x > -1.
2) 2x - 8 ≥ 7 - 3x
First, add 3x to both sides:
2x + 3x - 8 ≥ 7
Combine like terms:
5x - 8 ≥ 7
Next, add 8 to both sides:
5x - 8 + 8 ≥ 7 + 8
Simplify:
5x ≥ 15
Lastly, divide both sides by 5:
x ≥ 3
The solution to this equation is x ≥ 3.
3) 3(x + 2) < 14 - x
First, distribute the 3 on the left side of the equation:
3x + 6 < 14 - x
Next, add x to both sides:
3x + 6 + x < 14
Combine like terms:
4x + 6 < 14
Then, subtract 6 from both sides:
4x + 6 - 6 < 14 - 6
Simplify:
4x < 8
Lastly, divide both sides by 4:
x < 2
The solution to this equation is x < 2.
4) 4(x - 1) < x + 5
First, distribute the 4 on the left side of the equation:
4x - 4 < x + 5
Next, subtract x from both sides:
4x - x - 4 < 5
Combine like terms:
3x - 4 < 5
Then, add 4 to both sides:
3x - 4 + 4 < 5 + 4
Simplify:
3x < 9
Lastly, divide both sides by 3:
x < 3
The solution to this equation is x < 3.
5) 5 - 2(x + 1) > x
First, distribute the -2 on the left side of the equation:
5 - 2x - 2 > x
Next, simplify:
3 - 2x > x
Then, add 2x to both sides:
3 - 2x + 2x > x + 2x
Simplify:
3 > 3x
Lastly, divide both sides by 3:
1 > x
The solution to this equation is x < 1.
6) 6(x - 4) ≤ 2x
First, distribute the 6 on the left side of the equation:
6x - 24 ≤ 2x
Next, subtract 2x from both sides:
6x - 2x - 24 ≤ 0
Combine like terms:
4x - 24 ≤ 0
Then, add 24 to both sides:
4x - 24 + 24 ≤ 0 + 24
Simplify:
4x ≤ 24
Lastly, divide both sides by 4:
x ≤ 6
The solution to this equation is x ≤ 6.