Use the inequality to answer Parts 1-3.
-3(x-2)equal to or greater than 1/3
Part 1: Solve the inequality. Leave answer in terms of a whole number or reduced improper fraction.
Part 2: Write a verbal statement describing the solution to the inequality.
Part 3: Verify your solution to the inequality using two elements of the solution set.
To solve the inequality -3(x-2) ≥ 1/3, we can start by distributing the -3 to the terms inside the parentheses:
-3x + 6 ≥ 1/3
Next, we can subtract 6 from both sides to isolate the variable:
-3x ≥ 1/3 - 6
-3x ≥ -17/3
To get rid of the coefficient (-3) in front of x, we can divide both sides by -3. However, whenever we divide an inequality by a negative number, we must switch the direction of the inequality symbol. So, dividing both sides by -3 gives us:
x ≤ 17/9
Part 1: The solution to the inequality is x ≤ 17/9.
Part 2: The verbal statement describing the solution is "x is less than or equal to seventeen ninths."
Part 3: To verify the solution, we can substitute two elements of the solution set into the original inequality and see if they satisfy it. Let's choose x = 1 and x = 17/9 as our test values:
For x = 1:
-3(1-2) ≥ 1/3
-3(-1) ≥ 1/3
3 ≥ 1/3
This is true since 3 is indeed greater than or equal to 1/3.
For x = 17/9:
-3(17/9 - 2) ≥ 1/3
-3(17/9 - 18/9) ≥ 1/3
-3(-1/9) ≥ 1/3
1/3 ≥ 1/3
This is also true since 1/3 is equal to 1/3.
Both test values satisfy the original inequality, confirming that our solution x ≤ 17/9 is correct.
Part 1: To solve the inequality -3(x-2) ≥ 1/3, we can start by distributing -3 to (x-2):
-3(x-2) ≥ 1/3 becomes -3x + 6 ≥ 1/3
Next, we can subtract 6 from both sides:
-3x ≥ 1/3 - 6
Simplifying the right side:
-3x ≥ -17/3
Finally, we divide both sides by -3, but since we are dividing by a negative number, we need to reverse the inequality:
x ≤ -17/3 ÷ -3
Dividing gives us:
x ≤ 17/9
So, the solution to the inequality is x ≤ 17/9.
Part 2: A verbal statement describing the solution to the inequality is: "The solution to the inequality is any value of x less than or equal to 17/9."
Part 3: To verify the solution, we can choose two values of x that are less than or equal to 17/9 and substitute them into the original inequality to see if they satisfy it.
Let's choose x = 3 and x = 14/9.
For x = 3:
-3(3-2) = -3(1) = -3 ≤ 1/3
This is true since -3 is less than or equal to 1/3.
For x = 14/9:
-3(14/9-2) = -3(14/9-18/9) = -3(-4/9) = 12/3 = 4 ≥ 1/3
This is also true since 4 is greater than or equal to 1/3.
Both chosen values satisfy the original inequality, confirming that the solution is correct.
To solve the inequality -3(x-2) ≥ 1/3, we will apply the following steps:
Step 1: Distribute -3 to (x-2):
-3x + 6 ≥ 1/3
Step 2: Let's isolate the variable by subtracting 6 from both sides:
-3x ≥ 1/3 - 6
-3x ≥ 1/3 - 18/3
-3x ≥ -17/3
Step 3: Divide both sides of the inequality by -3. Since we are dealing with an inequality, we need to flip the direction of the inequality symbol when dividing by a negative number:
x ≤ (-17/3) / -3
x ≤ 17/9
Part 1: The solution to the inequality is x ≤ 17/9.
Part 2: A verbal statement describing the solution to the inequality is "x is less than or equal to seventeen ninths."
Part 3: To verify the solution, we can substitute two values from the solution set into the original inequality and check if the inequality holds true. Let's choose two values, one less than or equal to 17/9 and another greater than 17/9.
For x = 17/9 (which is on the boundary of the solution set):
-3(17/9 - 2) ≥ 1/3
-3(1/9) ≥ 1/3
-1/3 ≥ 1/3
Since -1/3 is not greater than or equal to 1/3, this value does not satisfy the inequality.
For x = 16/9 (which is less than 17/9):
-3(16/9 - 2) ≥ 1/3
-3(7/9) ≥ 1/3
-7/3 ≥ 1/3
Since -7/3 is greater than or equal to 1/3, this value satisfies the inequality.
Therefore, by verifying two values, we see that x ≤ 17/9 is the correct solution to the inequality.