Problem: Solve the system of inequalities \( 7x - 8 < 4x + 7 \) and \( -\frac{x}{3} > 9 \).
####
Step-by-Step Solution:
We are given two inequalities:
1. \( 7x - 8 < 4x + 7 \)
2. \( -\frac{x}{3} > 9 \)
We need to solve each inequality separately and then find the intersection of their solutions.
---
Solving Inequality 1: \( 7x - 8 < 4x + 7 \)
1. Start with the inequality:
\[
7x - 8 < 4x + 7
\]
2. Subtract \( 4x \) from both sides to isolate \( x \)-terms on one side:
\[
7x - 4x - 8 < 7
\]
\[
3x - 8 < 7
\]
3. Add 8 to both sides to isolate the \( x \)-term:
\[
3x - 8 + 8 < 7 + 8
\]
\[
3x < 15
\]
4. Divide both sides by 3:
\[
x < 5
\]
So, the solution to the first inequality is:
\[
x < 5
\]
---
Solving Inequality 2: \( -\frac{x}{3} > 9 \)
1. Start with the inequality:
\[
-\frac{x}{3} > 9
\]
2. Multiply both sides by 3 to eliminate the fraction. Note that multiplying by a positive number does not change the direction of the inequality:
\[
-x > 27
\]
3. Multiply both sides by \(-1\) to solve for \( x \). Remember that multiplying by a negative number reverses the inequality sign:
\[
x < -27
\]
So, the solution to the second inequality is:
\[
x < -27
\]
---
Finding the Intersection of Solutions
The solutions to the two inequalities are:
1. \( x < 5 \)
2. \( x < -27 \)
To find the intersection of these solutions, we take the more restrictive condition. Since \( x < -27 \) is stricter than \( x < 5 \), the intersection is:
\[
x < -27
\]
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Final Answer:
\[
\boxed{x < -27}
\]
Parent Tip: Review the logic above to help your child master the concept of one variable inequalities worksheet.
