Solve for x and graph the solution on the number line below: -34 ≤ -4x - 2 or -4x - 2 ≤ -46.
Understand the Problem
The question requires solving a set of inequalities involving the variable x and then graphing the solutions on a number line.
Answer
The solution is: $$ x \leq 8 \quad \text{or} \quad x \geq 11 $$
Answer for screen readers
The solution is:
$$ x \leq 8 \quad \text{or} \quad x \geq 11 $$
Steps to Solve
- Solve the first inequality
Start with the first inequality:
$$ -34 \leq -4x - 2 $$
Add 2 to both sides:
$$ -34 + 2 \leq -4x $$
This simplifies to:
$$ -32 \leq -4x $$
Now, divide both sides by -4, remembering to flip the inequality sign:
$$ \frac{-32}{-4} \geq x $$
This gives us:
$$ 8 \geq x $$
or
$$ x \leq 8 $$
- Solve the second inequality
Now solve the second inequality:
$$ -4x - 2 \leq -46 $$
Add 2 to both sides:
$$ -4x \leq -46 + 2 $$
This simplifies to:
$$ -4x \leq -44 $$
Next, divide both sides by -4 and flip the inequality sign:
$$ x \geq \frac{-44}{-4} $$
This results in:
$$ x \geq 11 $$
- Combine the solutions
Now we combine the results from both inequalities.
The solutions are:
$$ x \leq 8 \quad \text{or} \quad x \geq 11 $$
- Graph the solution on the number line
To graph this, mark the points 8 and 11 on the number line.
- Use a closed circle on 8 and shade to the left.
- Use a closed circle on 11 and shade to the right.
The solution is:
$$ x \leq 8 \quad \text{or} \quad x \geq 11 $$
More Information
The solution represents two intervals on the number line. The inequalities show that any value of ( x ) less than or equal to 8, or greater than or equal to 11, satisfies the original conditions. This means that the solution is not continuous; instead, it creates gaps between the solutions.
Tips
- Forgetting to flip the inequality sign when dividing by a negative number. Always remember to flip the sign when doing this.
- Misplacing the values on the number line; careful attention to the shading of intervals is necessary.
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