Solve for x and graph the solution on the number line below. -x - 5 ≥ 1 or x - 5 < -8.
Understand the Problem
The question is asking to solve the inequalities for x and then plot the solutions on a number line. The inequalities are -x - 5 ≥ 1, and x - 5 < -8.
Answer
$$ x \leq -3 $$
Answer for screen readers
The solution is expressed as:
$$ x \leq -3 $$
Steps to Solve
- Solve the first inequality
Start with the inequality:
$$ -x - 5 \geq 1 $$
Add 5 to both sides:
$$ -x \geq 6 $$
Now, multiply by -1 (remember to flip the inequality sign):
$$ x \leq -6 $$
- Solve the second inequality
Now, tackle the second inequality:
$$ x - 5 < -8 $$
Add 5 to both sides:
$$ x < -3 $$
- Combine the solutions
The combined solutions from both inequalities are:
$$ x \leq -6 \text{ or } x < -3 $$
Since $x \leq -6$ overlaps with $x < -3$, we can summarize the solution as:
$$ x \leq -3 $$
- Plot the solutions on a number line
On the number line, represent the solution:
- For $x \leq -6$, use a closed circle at -6 and shade to the left.
- For $x < -3$, use an open circle at -3 and shade to the left.
This combines to show all solutions to the left of -3.
The solution is expressed as:
$$ x \leq -3 $$
More Information
This problem involves solving linear inequalities and understanding how to represent solutions graphically. The solution to each inequality was determined separately, and consideration of overlapping intervals was crucial for the final answer.
Tips
- Not flipping the inequality sign when multiplying or dividing by a negative number. Always remember to flip the sign.
- Failing to properly graph the solutions. Ensure to use closed circles for ≤ and ≥, and open circles for < and >, and to shade appropriately.
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