Question

Using the properties of inequality, justify both steps in solving the inequality 25m−7≤17
.(4 points)

Answer 1

in one short thing

Answer 2

To solve the inequality \( 25m - 7 \leq 17 \), we can break it down into two steps and justify each step using properties of inequalities.

**Step 1: Add 7 to both sides**

We start by isolating the term involving \( m \). We add 7 to both sides of the inequality:

\[
25m - 7 + 7 \leq 17 + 7
\]

This simplifies to:

\[
25m \leq 24
\]

**Justification:** Adding the same number (in this case, 7) to both sides of an inequality does not change the direction of the inequality. This is a property of inequalities that states: if \( a \leq b \), then \( a + c \leq b + c \) for any real number \( c \).

---

**Step 2: Divide both sides by 25**

Next, we divide both sides of the inequality by 25:

\[
\frac{25m}{25} \leq \frac{24}{25}
\]

This simplifies to:

\[
m \leq \frac{24}{25}
\]

**Justification:** Dividing both sides of an inequality by a positive number (in this case, 25) does not change the direction of the inequality. This is another property of inequalities which states: if \( a \leq b \) and \( c > 0 \), then \( \frac{a}{c} \leq \frac{b}{c} \).

---

Combining both steps, the solution to the inequality \( 25m - 7 \leq 17 \) is:

\[
m \leq \frac{24}{25}
\]

Answer 3

To solve the inequality \( 25m - 7 \leq 17 \):

1. **Add 7 to both sides:**
\[
25m - 7 + 7 \leq 17 + 7 \implies 25m \leq 24
\]
(Justification: Adding the same number to both sides preserves the inequality.)

2. **Divide both sides by 25:**
\[
\frac{25m}{25} \leq \frac{24}{25} \implies m \leq \frac{24}{25}
\]
(Justification: Dividing both sides by a positive number preserves the inequality.)

Thus, the solution is \( m \leq \frac{24}{25} \).