To find the solution set for the inequality \(-2x + 5 + x \le 3x + 13\), we first simplify the inequality:
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Combine like terms on the left-hand side: \[ -2x + x + 5 \le 3x + 13 \] This simplifies to: \[ -x + 5 \le 3x + 13 \]
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Next, we move all \(x\) terms to one side and constant terms to the other side. Adding \(x\) to both sides gives: \[ 5 \le 4x + 13 \]
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Then, subtract \(13\) from both sides: \[ 5 - 13 \le 4x \] Simplifying this results in: \[ -8 \le 4x \]
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To isolate \(x\), divide both sides by \(4\): \[ -2 \le x \] or equivalently, \[ x \ge -2 \]
The solution set is \(x \ge -2\).
Graphical Representation
To represent this inequality graphically on a number line:
- The number \(-2\) will be marked on the number line.
- Since the inequality is \(x \ge -2\), we will use a closed dot (or filled circle) at \(-2\) to indicate that \(-2\) is included in the solution set.
- The region to the right of \(-2\) will be shaded to represent all the \(x\)-values greater than \(-2\).
Look for a graph that shows a closed dot at \(-2\) and shading to the right. This will best represent the solution set for the inequality \(x \ge -2\).
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