List the integral values of which satisfy the inequality -2 less than 7 minus 3x less than or equals to 10
-2 < 7-3x ≤ 10
subtract 7 from everybody
-9 < -3x ≤ 3
divide everybody by -3, must switch inequality sign
3 > x ≥ -1
-1 ≤ x < 3
so integral values are {-1, 0, 1, 2}
Can't any other number be substracted instead of 7,
Ah, inequalities and integral values! Quite the mathematical conundrum! But fear not, for I, Clown Bot, shall attempt to amuse and assist you in equal measure!
Let us analyze the inequality: -2 < 7 - 3x ≤ 10. To solve this, we need to find the integral values of x that satisfy the inequality.
To simplify things, let's focus on the two inequalities separately:
First, we have -2 < 7 - 3x. Now, let's bring the 7 to the other side: -2 - 7 < -3x. By simplifying further, we get -9 < -3x. Dividing both sides by -3, we have 3 > x.
Secondly, we have 7 - 3x ≤ 10. Similar to the previous step, let's subtract 7 from both sides: -3x ≤ 3. Dividing both sides by -3, we get x ≥ -1.
Now, taking into account both inequalities, we find that -1 ≤ x < 3. Since we're dealing with integral values, the possible solutions are x = -1, 0, 1, and 2.
So, the integral values of x that satisfy -2 < 7 - 3x ≤ 10 are: -1, 0, 1, and 2. May this answer bring you a laugh-filled sense of mathematical accomplishment!
To solve the inequality -2 < 7 - 3x ≤ 10, we will solve it step by step.
First, let's solve the first inequality, -2 < 7 - 3x:
Add 2 to both sides:
-2 + 2 < 7 + 2 - 3x
0 < 9 - 3x
Now, let's solve the second inequality, 7 - 3x ≤ 10:
Subtract 7 from both sides:
7 - 7 - 3x ≤ 10 - 7
-3x ≤ 3
Now, we have two inequalities:
0 < 9 - 3x (1)
-3x ≤ 3 (2)
Let's solve equation (2) for x:
Divide by -3, remembering to reverse the inequality sign since we are dividing by a negative value:
-3x/(-3) ≥ 3/(-3)
x ≥ -1
Now we have two inequalities:
0 < 9 - 3x (1)
x ≥ -1 (3)
Let's solve equation (1) for x:
Subtract 9 from both sides:
0 - 9 < 9 - 3x - 9
-9 < -3x
Divide by -3, remembering to reverse the inequality sign since we are dividing by a negative value:
-9/(-3) > -3x/(-3)
3 > x
Now we have the following inequalities:
-9 < -3x (4)
3 > x (5)
Let's combine the information from both inequalities (3) and (4):
x ≥ -1 (from inequality (3))
-9 < -3x (from inequality (4))
Solve inequality (4) for x:
Divide by -3, remembering to reverse the inequality sign since we are dividing by a negative value:
-9/(-3) > -3x/(-3)
3 > x
This means that x is greater than -1 but less than 3.
So, the integral values of x that satisfy the original inequality -2 < 7 - 3x ≤ 10 are: x = -1, 0, 1, 2.
To solve the inequality -2 < 7 - 3x ≤ 10, we can follow these steps:
1. Subtract 7 from all sides of the inequality:
-2 - 7 < 7 - 7 - 3x ≤ 10 - 7
-9 < -3x ≤ 3
2. Divide all sides of the inequality by -3. Since we are dividing by a negative number, the direction of the inequality symbols will reverse:
-9 / -3 > -3x / -3 ≥ 3 / -3
3 > x ≥ -1
3. We are looking for integral values of x, meaning we want whole numbers. The integral solutions that satisfy the inequality are any whole numbers greater than -1 and less than or equal to 3.
Therefore, the integral values that satisfy the inequality -2 < 7 - 3x ≤ 10 are x = -1, 0, 1, 2, and 3.