Question 937·Hard·Math

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|x-3| = \sqrt{2x-1}

What is the solution set of the equation above?

An absolute value measures distance, while a square root is never negative. When both sides of an equation change with $x$ , graph the original sides in Desmos and click every intersection. Then compare the intersections’ $x$ -coordinates with the exact expressions in the choices. A zero of one side alone does not make the two sides equal.

Hints

Hint 1
An intersection is where two graphs have the same height. Graph each original side of the equation separately. How many times do they meet, and which coordinate of each point gives a solution for $x$ ?
Hint 2
A solution set contains every value that works. The absolute-value side becomes $0$ when $x=3$ , but that is not enough. What does the square-root side equal at that input?
Hint 3
Desmos shows intersection coordinates as rounded decimals, while the choices use exact square roots. Evaluate the exact expressions in the choices separately, then compare each result with an intersection’s $x$ -coordinate.

Step-by-step

Graph the original sides
Step 1
Find where the sides are equal
Type $y=|x-3|$ and $y=\sqrt{2x-1}$ on separate lines. Click both intersections. Desmos shows $x\approx1.551$ and $x\approx6.449$ . At an intersection, the two sides have the same value, so each $x$ -coordinate solves the equation as written.
Step 2
Check the tempting extra value
At $x=3$ , the absolute-value side is $0$ . Type each original side with $3$ substituted: Desmos gives $0$ and about $2.236$ , respectively. They are unequal, so $3$ is not a solution, even though it makes one side zero.
Step 3
Match both intersections to exact values
Type $4-\sqrt6$ and $4+\sqrt6$ . Desmos prints about $1.551$ and $6.449$ , matching the two intersection $x$ -coordinates. The solution set is $\{4-\sqrt6,\ 4+\sqrt6\}$ . Choice C.

Question 937·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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