The largest real value of a for which the equation $|x + a| + |x - 1| = 2$ has an infinite number of solutions for $x$ is

We need to find the largest real value of $a$ for which $|x + a| + |x - 1| = 2$ has infinitely many solutions.

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The expression $|x + a| + |x - 1|$ represents the sum of two distances:

$|x + a| = |x - (-a)|$ = distance from point $x$ to point $-a$

$|x - 1|$ = distance from point $x$ to point $1$

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For any two points $p$ and $q$ on a number line, the sum $|x - p| + |x - q|$ has:

Minimum value $= |q - p|$ (the distance between the two points)

This minimum is achieved when $x$ lies anywhere between $p$ and $q$

If $x$ is outside this interval, the sum is greater than $|q - p|$

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In our case, the two fixed points are $-a$ and $1$.

The sum $|x + a| + |x - 1|$ has:

Minimum value $= |1 - (-a)| = |1 + a|$

This minimum occurs when $x$ lies between $-a$ and $1$

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For the equation $|x + a| + |x - 1| = 2$ to have infinitely many solutions, the constant value $2$ must equal the minimum possible value of the left side.

Therefore: $|1 + a| = 2$

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$|1 + a| = 2$ means:

$1 + a = 2$ or $1 + a = -2$

$a = 1$ or $a = -3$

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The largest value is $a = 1$.

When $a = 1$, our equation becomes $|x + 1| + |x - 1| = 2$.

For any $x$ between $-1$ and $1$: $|x + 1| + |x - 1| = (x + 1) + (1 - x) = 2$

This gives infinitely many solutions (all $x$ in $[-1, 1]$).

Answer: $a = 1$