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In this section, you will solve square root inequalities containing a numerical value instead of y.

You must compare both sides of the inequality.

Example: Solve 2√x ≤ 2 and 2√x ≥ 2

Step 1: Both ≤ and ≥ are solid lines.

Step 2: Graph both sides of the inequality as functions on the same coordinate plane.

Graph y = 2√x and y = 2.

graph of y = 2 times square root of x and y = 2

Look for any points of intersection of the two graphs, where the two graphs are equal.
The intersection point is included as part of the solution if the inequality symbol is ≤ or ≥ and will be a closed point on the graph.

Step 3: Find where the function y = √2x is greater than y = 2 and less than y = 2.

graph of y = 2 times square root of x and y = 2 with green shading above the line y = 2 and yellow shading below the line y = 2

Identify the x-value, x = 1, of the point of intersection. It is the change-over point between below y = 2 (less than) and above y = 2 (greater than).

Going back to our original example: √x ≤ 2 and √2 ≥ 2

graph of y = 2 times square root of x and y = 2 with yellow shading below the line y = 2 and the portion of the graph of y = 2 times sqaure root of x between y = 0 and y = 2 indicated in red.

If you are graphing √2x ≤ 2, then you want the less than part which is below
y = 2, the yellow area.

Because the original inequality only contained an x-variable, then the solution only has x’s.

The solution contains any x-values that produce a y-value less than or equal to 2.

The solution to the graph of √2x ≤ 2 is 0 ≤ x ≤ 1.

graph of y = 2 times square root of x and y = 2 with the portion of the graph of y = 2 times square root of x between y = 0 and y = 2 indicated in red

graph of y = 2 times square root of x and y = 2 with green shading above the line y = 2 and the portion of the graph of y = 2 times sqaure root of x greater than y = 2 indicated in red.

If you are graphing √2x ≥ 2, then you want the greater than part which is above y = 2, the green area.

The solution contains any x-values that produce a y-value greater than or equal to 2.

The solution to the graph of √2x ≥ 2 is x ≥ 1.

graph of y = 2 times sqaure root of x and y = 2 with the portion of the graph of y = 2 times sqaure root of x greater than y = 2 indicated in red.

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