To determine which of the provided equations has no x-intercepts, we need to analyze the equations based on their definitions. A graph has an x-intercept at points where the y-coordinate equals zero (i.e., when the equation equals zero).
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−|x| + 3: Set this equal to zero: \[ -|x| + 3 = 0 \implies |x| = 3 \] This has solutions \(x = 3\) and \(x = -3\), so it has x-intercepts.
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|x| − 3: Set this equal to zero: \[ |x| - 3 = 0 \implies |x| = 3 \] This also has solutions \(x = 3\) and \(x = -3\), so it has x-intercepts.
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|x| + 3: Set this equal to zero: \[ |x| + 3 = 0 \] Since the absolute value function is always non-negative, this equation has no solutions. Therefore, the graph of this equation does NOT have any x-intercepts.
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−|x| − 3: Set this equal to zero: \[ -|x| - 3 = 0 \implies -|x| = 3 \] This equation has no solutions since the left side can't be positive. Therefore, this graph also does NOT have any x-intercepts.
In conclusion, the equations that have no x-intercepts are |x| + 3 and −|x| − 3. However, the most straightforward example meeting the criteria of having "no x-intercepts" due to its structure is:
|x| + 3 (absolute value of x plus 3)