To justify both steps in solving the inequality ((2/5)m - 7) ≤ 17, let's break it down into two separate steps:
Step 1: Isolate the variable term.
In this step, we want to isolate the variable term, which is (2/5)m, on one side of the inequality. To do this, we need to get rid of the constant term (-7) by adding 7 to both sides of the inequality:
((2/5)m - 7) + 7 ≤ 17 + 7
Simplifying the equation,
(2/5)m ≤ 24
Step 2: Solve for the variable.
In this step, we want to solve for the variable term, which is (2/5)m. To do this, we need to get rid of the coefficient (2/5) by multiplying both sides of the inequality by the reciprocal of 2/5, which is 5/2. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality symbol. Since 5/2 is positive, we don't need to worry about reversing the direction:
((2/5)m) * (5/2) ≤ 24 * (5/2)
Simplifying the equation,
m ≤ 60/2
m ≤ 30
Therefore, the inequality (2/5)m - 7 ≤ 17 is satisfied when m is less than or equal to 30.
To justify these steps using numbers, let's substitute a value for m and see if the inequality holds:
Let's take m = 30:
(2/5)(30) - 7 ≤ 17
12 - 7 ≤ 17
5 ≤ 17
The inequality holds.
Now, let's take m = 31:
(2/5)(31) - 7 ≤ 17
12.4 - 7 ≤ 17
5.4 ≤ 17
Again, the inequality holds.
By substituting different values for m, we can see that the inequality (2/5)m - 7 ≤ 17 holds true for m ≤ 30. Hence, these steps are justified.