using the properties of inquality, justify both steps in solving the inequality 2/5m - 7 <_ 17

Step 1: Isolate the variable term

To solve the inequality, the first step is to isolate the variable term on one side of the inequality. In this case, that means getting rid of the -7 on the left side. We can do this by adding 7 to both sides of the inequality:

(2/5)m - 7 + 7 ≤ 17 + 7

Simplifying, we get:

(2/5)m ≤ 24

Step 2: Multiply both sides by the reciprocal

The second step is to multiply both sides of the inequality by the reciprocal of the fraction coefficient (2/5). In this case, the reciprocal is 5/2. By multiplying both sides by 5/2, we can eliminate the fraction:

(5/2)(2/5)m ≤ (5/2)(24)

Simplifying, we get:

(1)m ≤ 60/2
m ≤ 30

So, the solution to the inequality is m ≤ 30.

rewrite the equation 4x/3x-12 + 3 = 10 in two equations joined by " and."

Sure! Here are the two equations:

1. 4x/(3x - 12) + 3 = 10

Equation 1 can be rewritten as:

4x/(3x - 12) = 7

and

2. 4x/(3x - 12) + 3 = 10

Equation 2 can be rewritten as:

4x/(3x - 12) = 7

To solve the inequality 2/5m - 7 ≤ 17 step-by-step using the properties of inequalities, we need to isolate the variable m.

Step 1: Add 7 to both sides
2/5m - 7 + 7 ≤ 17 + 7
2/5m ≤ 24

Adding 7 to both sides of the inequality helps to eliminate the -7 on the left side and move the constant term to the right side.

Step 2: Multiply both sides by 5/2
(5/2)(2/5m) ≤ (5/2)(24)
1m ≤ 60/2
1m ≤ 30

To eliminate the fraction coefficient 2/5 in front of m, we multiply both sides of the inequality by its reciprocal 5/2. This step is valid since we are applying the multiplication property of inequalities that states that if a < b and c > 0, then ac < bc.

Therefore, the solution to the inequality 2/5m - 7 ≤ 17 is m ≤ 30.

To solve the inequality 2/5m - 7 ≤ 17, we will use the properties of inequalities. Here are the steps along with their justifications:

Step 1: Add 7 to both sides of the inequality:
(2/5)m - 7 + 7 ≤ 17 + 7
(2/5)m ≤ 24

Justification: According to the addition property of inequalities, if we add the same value to both sides of an inequality, the inequality remains unchanged.

Step 2: Multiply both sides by 5/2 (or 2/5 reciprocal):
(2/5)m * (5/2) ≤ 24 * (5/2)
m/1 ≤ 120/2
m/1 ≤ 60

Justification: According to the multiplication property of inequalities, if we multiply both sides of an inequality by a positive number, the direction of the inequality remains the same. In this case, we multiplied by 5/2 (or its reciprocal 2/5), which is positive.

Therefore, the solution to the inequality 2/5m - 7 ≤ 17 is m ≤ 60.