Solve the inequality. Graph the solution set and write it in interval notation.
- 3(x - 2) - 2x < - (4x + 6) + 3x
Write the answer in interval notation.
__ (Type your answer in interval notation.)
Let's simplify the equation step by step:
-3(x - 2) - 2x < - (4x + 6) + 3x
Distribute the -3 and the -1 on the right side:
-3x + 6 - 2x < -4x - 6 + 3x
Combine like terms:
-5x + 6 < -x - 6
Add 5x to both sides:
6 < 4x - 6
Add 6 to both sides:
12 < 4x
Divide both sides by 4 (and flip the sign because we are dividing by a negative number):
3 > x
So, the solution to the inequality is x < 3.
The graph of this inequality would be an open circle at 3 on the number line, and shading to the left.
In interval notation, the solution set can be written as (-∞, 3).
To solve the inequality, let's simplify the expression:
-3(x - 2) - 2x < - (4x + 6) + 3x
Step 1: Distribute the -3 and the -1 to each term inside the parentheses on the left side:
-3x + 6 - 2x < -4x - 6 + 3x
Step 2: Combine like terms on both sides of the inequality:
-5x + 6 < -x - 6
Step 3: Add 5x to both sides to isolate the variable:
6 < 4x - 6
Step 4: Add 6 to both sides:
12 < 4x
Step 5: Divide both sides by 4 (note: since we are dividing by a positive number, the inequality sign doesn't change):
3 < x
Now let's graph the solution set on a number line:
o--------------------------------
-3 -2 -1 0 1 2 3 4 5 6 7 8
The solution set is all values of x that are greater than 3.
In interval notation, this is written as:
(3, ∞)
To solve the inequality -3(x - 2) - 2x < -(4x + 6) + 3x, we'll start by simplifying and combining like terms on both sides.
First, distribute -3 to the terms in the parentheses on the left side and distribute -1 to the terms in parentheses on the right side:
- 3x + 6 - 2x < -4x - 6 + 3x
Simplifying this further:
-5x + 6 < -x - 6
Next, move all the x terms to one side and the constant terms to the other side by adding 5x to both sides and subtracting 6 from both sides:
-5x + 5x + 6 + 6 < -x - 5x - 6 - 6
Simplifying:
6 < -6x - 12
Now, isolate the x term by moving the constant terms to the other side. In this case, add 6x to both sides:
6 + 6x < -12
Finally, subtract 6 from both sides:
6x < -18
To solve for x, divide both sides of the inequality by 6:
x < -18/6
Simplifying:
x < -3
So, the solution to the inequality is x < -3.
To graph the solution set on a number line, draw an open circle at -3 and shade the line to the left of -3. This represents all the values of x that satisfy the inequality.
In interval notation, we write the solution set as (-∞, -3).