1. 2x > -6 and x - 4 < 3
2. x + 3 > 2x + 1 and -4x < -8
3. -6 < x + 3 < 6
4. -3x > -6 or x - 5 < 2
5. x - 2 > 2x + 1 or 10 > -2x + 2
LOL - why me ?
1. 2x > -6 and x - 4 < 3
x > -3 and x < +7
-3 < x < +7
2. x + 3 > 2x + 1 and -4x < -8
note if you divide by negative number, change arrow direction
2 > x and x > 2 can not be both
3. -6 < x + 3 < 6
subtract 3 from everything
-9 < x < 3
'cause you rock (: !
thanks for helping me so far (:
4. -3x > -6 or x - 5 < 2
x < 2
or x < 7
everything left of seven satisfies the second, everything left of 2 satisfies both. In other words x<2 is a subset of x <7
5. x - 2 > 2x + 1 or 10 > -2x + 2
x < -2 or x > -4
that is everywhere
To solve these inequalities, we will use a combination of algebraic manipulation and logic operations. Let's go through each inequality step by step:
1. 2x > -6 and x - 4 < 3
- To solve the first inequality, divide both sides by 2: x > -3.
- For the second inequality, add 4 to both sides: x < 7.
- The solution to the system of inequalities is -3 < x < 7.
2. x + 3 > 2x + 1 and -4x < -8
- Start by subtracting x from both sides of the first inequality: 3 > x + 1.
- Subtract 1 from both sides: 2 > x.
- For the second inequality, divide both sides by -4. Since we are dividing by a negative number, the inequality sign flips: x > 2.
- However, we also need to consider the intersection of these two inequalities. The solution is 2 < x < 3.
3. -6 < x + 3 < 6
- Begin by subtracting 3 from all sides of the inequality: -9 < x < 3.
- This means that x must be greater than -9 and less than 3.
4. -3x > -6 or x - 5 < 2
- For the first inequality, divide both sides by -3. Since we are dividing by a negative number, the inequality sign flips: x < 2.
- The second inequality states that x - 5 < 2. Add 5 to both sides: x < 7.
- The solution to this system of inequalities is x < 2 or x < 7.
5. x - 2 > 2x + 1 or 10 > -2x + 2
- Begin by simplifying the first inequality: -1 > x.
- For the second inequality, subtract 2 from both sides: 8 > -2x.
- Divide both sides by -2. Since we are dividing by a negative number, the inequality sign flips: x > -4.
- The solution is x > -4 or -1 > x.
Remember to always double-check your work and the logic behind the steps in solving inequalities.