Question
Using the properties of inequality, justify both steps in solving the inequality 25m−8<−2
.
25m−8<−2
25m−8+8<−2+8
52(25m)<52(6)
Question 2
Solve for m
$$
.
25m−8<−2
25m−8+8<−2+8
52(25m)<52(6)
Question 2
Solve for m
$$
Answers
GPT-4o mini
To solve the inequality \( 25m - 8 < -2 \), we proceed step by step while justifying each step based on the properties of inequalities.
### Step 1: Isolate the term containing \( m \)
We start with the original inequality:
\[
25m - 8 < -2
\]
To isolate \( 25m \), we add \( 8 \) to both sides of the inequality. This step is justified by the property of inequalities that states if \( a < b \), then \( a + c < b + c \) for any real number \( c \).
\[
25m - 8 + 8 < -2 + 8
\]
This simplifies to:
\[
25m < 6
\]
### Step 2: Solve for \( m \)
Next, we want to solve for \( m \). To do this, we divide both sides of the inequality by \( 25 \). This is justified by the property of inequalities that states if \( a < b \) and \( c > 0 \), then \( \frac{a}{c} < \frac{b}{c} \).
\[
\frac{25m}{25} < \frac{6}{25}
\]
This simplifies to:
\[
m < \frac{6}{25}
\]
### Final Solution
Thus, the solution to the inequality is:
\[
m < \frac{6}{25}
\]
In conclusion, both steps involve properties of inequalities, namely the addition property (adding the same value to both sides) and the division property (dividing both sides by a positive number). The final answer is \( m < \frac{6}{25} \).
### Step 1: Isolate the term containing \( m \)
We start with the original inequality:
\[
25m - 8 < -2
\]
To isolate \( 25m \), we add \( 8 \) to both sides of the inequality. This step is justified by the property of inequalities that states if \( a < b \), then \( a + c < b + c \) for any real number \( c \).
\[
25m - 8 + 8 < -2 + 8
\]
This simplifies to:
\[
25m < 6
\]
### Step 2: Solve for \( m \)
Next, we want to solve for \( m \). To do this, we divide both sides of the inequality by \( 25 \). This is justified by the property of inequalities that states if \( a < b \) and \( c > 0 \), then \( \frac{a}{c} < \frac{b}{c} \).
\[
\frac{25m}{25} < \frac{6}{25}
\]
This simplifies to:
\[
m < \frac{6}{25}
\]
### Final Solution
Thus, the solution to the inequality is:
\[
m < \frac{6}{25}
\]
In conclusion, both steps involve properties of inequalities, namely the addition property (adding the same value to both sides) and the division property (dividing both sides by a positive number). The final answer is \( m < \frac{6}{25} \).