Fill in the compound inequality
4(x-6) 2(x-10) and 5(x + 2) >/ 2(x +8)
with less than, greater than, at least, or at most so that the solution is only one value
Well, let's "solve" this inequality with a little bit of clown humor:
4(x-6) < 2(x-10) and 5(x + 2) > 2(x +8)
Let's start with the first inequality 4(x-6) < 2(x-10). We need to make it a little funnier, so let's say it's "less than a shy clown trying to get on stage":
4(x-6) < 2(x-10) becomes 4(x-6) < 2(x-10) < Shy Clown
Now, for the second inequality 5(x + 2) > 2(x + 8). This time, let's make it "greater than a clown juggling flaming torches":
5(x + 2) > 2(x + 8) becomes Juggler Clown > 2(x + 8)
And finally, we want to combine the two inequalities into one. To make it even more entertaining, let's imagine it as a clown balancing on a tightrope:
Shy Clown < 4(x-6) < 2(x-10) < Juggler Clown > 2(x + 8)
Voilà! The solution is now in one value - that funny tightrope-walking clown! But jokes aside, remember to solve the inequalities step by step to find the actual solution.
To fill in the compound inequality with the correct symbols, we need to determine whether the given inequality is true for exactly one value or more than one value.
Let's simplify the expression step-by-step and determine the correct symbols.
1. Simplify the left side of the inequality:
4(x - 6) - 2(x - 10)
Distribute the terms:
4x - 24 - 2x + 20
Combine like terms:
2x - 4
2. Simplify the right side of the inequality:
5(x + 2) - 2(x + 8)
Distribute the terms:
5x + 10 - 2x - 16
Combine like terms:
3x - 6
Now, the original inequality becomes:
2x - 4 ≥ 3x - 6
To determine the correct symbol, we need to solve this inequality for x. Subtract 2x from both sides:
-4 ≥ x - 6
Add 6 to both sides:
2 ≥ x
So, the solution for x is x ≤ 2.
Therefore, the compound inequality should be written as:
2 ≥ x ≤ 2 or at most x ≤ 2.
To fill in the compound inequality with either "less than," "greater than," "at least," or "at most" in order to have a single solution, we need to compare the expressions on both sides.
Starting with the left side of the inequality, let's simplify it:
4(x - 6) = 4x - 24
And for the right side of the inequality:
2(x - 10) = 2x - 20
Now, let's simplify the inequality further:
5(x + 2) ≥ 2(x + 8)
Distribute the values inside the brackets:
5x + 10 ≥ 2x + 16
Next, move all the x terms to one side and the constant terms to the other side:
5x - 2x ≥ 16 - 10
Combine like terms:
3x ≥ 6
Now, to solve for x, divide both sides of the inequality by 3:
x ≥ 2
So, we have x being "at least" or "greater than or equal to" 2.
Revisiting the compound inequality we need to fill:
4(x - 6) ≤ 2(x - 10) and 5(x + 2) ≥ 2(x + 8)
Substituting x ≥ 2, we have:
4(2 - 6) ≤ 2(2 - 10) and 5(2 + 2) ≥ 2(2 + 8)
Calculating both sides:
4(-4) ≤ 2(-8) and 5(4) ≥ 2(10)
-16 ≤ -16 and 20 ≥ 20
Both inequalities are true, so x being at least 2 satisfies the compound inequality and provides a single solution.