Problem of the Day


For what values of x is |x-4|-|x-3|\geq7?

A. x\leq3
B. 3\leq x\leq4
C. x\geq4
D. x\geq7
E. No values of x satisfy this inequality.

Reveal Answer

Answer

[latexpage]

E. No values of $x$ satisfy this inequality.

See the Solution

Solution

[latexpage]

For what values of $x$ is $|x-4|-|x-3|\geq7$?

Algebraic expressions with absolute value signs are difficult to work with, so one approach is to¬†rewrite the expression in an equivalent way that does not require absolute value signs. To do this, we’ll need to break the problem into three cases: $x\leq3$, $3<x<4$, and $x\geq4$.

Case 1: $x\leq3$

If we are only considering values of $x$ that are less than or equal to 3, then we can rewrite our expression as $(4-x)-(3-x)\geq7$. This simplifies to $1\geq7$, which is definitely not true. This tells us that there is no solution to this inequality where $x\leq3$.

Case 2: $3<x<4$

In this case we can rewrite the expression as $(4-x)-(x-3)\geq7$. This simplifies to $-2x+7\geq7$ or $-2x\geq0$ or $x\leq0$. This may appear to provide a solution set, but keep in mind that the way we rewrote the expression in this case is ONLY valid if $x$ is between 3 and 4. None of the values in $x\leq0$ are between 3 and 4, so we cannot include any of them in our solution set based on the work from this case. So, no $x$ values between 3 and 4 solve this inequality.

Case 3: $x\geq4$

In this case we can rewrite the expression as $(x-4)-(x-3)\geq7$ which simplifies to $-7\geq7$. This is not true, which means there is no solution to this inequality where $x\geq4$.

All three of our cases failed to produce a solution, and all real numbers were accounted for between all three cases. Therefore, no solution to this inequality exists.

 


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