Solve for x and graph the solution on the number line: -2 ≤ -x + 4 < 3.
Understand the Problem
The question is asking to solve the inequality for x and then graph the solution on a number line. The inequality is presented as -2 ≤ -x + 4 < 3. This means we need to isolate x to find its range and illustrate the solution graphically.
Answer
The solution is \( 1 < x \leq 6 \).
Answer for screen readers
The solution to the inequality is ( 1 < x \leq 6 ).
Steps to Solve
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Separate the inequalities We start by separating the compound inequality into two parts:
$$ -2 \leq -x + 4 $$
and
$$ -x + 4 < 3 $$ -
Solve the first inequality For the first inequality, isolate $x$:
$$ -2 \leq -x + 4 $$
Subtract 4 from both sides:
$$ -2 - 4 \leq -x $$
$$ -6 \leq -x $$
Multiply by -1 (remember to reverse the inequality sign):
$$ 6 \geq x $$
or
$$ x \leq 6 $$ -
Solve the second inequality Now solve the second part:
$$ -x + 4 < 3 $$
Subtract 4 from both sides:
$$ -x < 3 - 4 $$
$$ -x < -1 $$
Multiply by -1 (reverse the inequality sign):
$$ x > 1 $$ -
Combine the results Now combine the two inequalities:
$$ 1 < x \leq 6 $$ -
Graph the solution To graph this on a number line, use an open circle on 1 (to indicate that 1 is not included) and a closed circle on 6 (to indicate that 6 is included). Draw a line between these points.
The solution to the inequality is ( 1 < x \leq 6 ).
More Information
This solution shows that ( x ) can take any value greater than 1 up to and including 6. Graphically, this means the region between these two values is shaded on the number line.
Tips
- Forgetting to reverse the inequality sign when multiplying by a negative number.
- Not correctly representing open and closed circles on the number line for inequalities.
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