Solve for x and graph the solution on the number line below. 17 < x + 10 and 19 ≥ x + 10.
Understand the Problem
The question is asking to solve for x in two inequalities and graph the solution on a number line. The inequalities provided are 17 < x + 10 and 19 ≥ x + 10.
Answer
The interval for the solution is $(7, 9]$.
Answer for screen readers
The solution for the inequalities is expressed in interval notation as: $$ (7, 9] $$
Steps to Solve
- Solve the first inequality
Start with the first inequality: $$ 17 < x + 10 $$
To isolate $x$, subtract 10 from both sides: $$ 17 - 10 < x $$ $$ 7 < x $$
This can be rewritten as: $$ x > 7 $$
- Solve the second inequality
Now solve the second inequality: $$ 19 \geq x + 10 $$
Again, subtract 10 from both sides: $$ 19 - 10 \geq x $$ $$ 9 \geq x $$
This can be rewritten as: $$ x \leq 9 $$
- Combine the results
Now combine the results from both inequalities: $$ 7 < x \leq 9 $$
This indicates that $x$ is greater than 7 and less than or equal to 9.
- Graph the solution on the number line
To graph the solution $7 < x \leq 9$:
- Use an open circle at 7 (indicating that 7 is not included).
- Use a closed circle at 9 (indicating that 9 is included).
- Shade the region between 7 and 9.
The solution for the inequalities is expressed in interval notation as: $$ (7, 9] $$
More Information
The solution indicates that $x$ must be greater than 7 but can be equal to 9. Graphing involves marking open and closed circles appropriately, which visually communicates the boundaries of the solution set.
Tips
- Forgetting to switch the inequality direction when multiplying or dividing by a negative number (not applicable here since we only add/subtract).
- Misplacing the open and closed circles when graphing the solution.
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