In this section, you will be solving square root inequalities by graphing and using tables. You will need a calculator for this section.

Steps to Graph Square Root Inequalities

Text: Steps to Graph Square Root Inequalities Step 1 Decide whether the curve should be dotted or solid. Use dotted for < or > and solid for 'less than or equal to' or 'greater than or equal to' Step 2 Graph the equation. Step 3 Choose a test point that lies above or below the graph When choosing your test point for square root functions, try to pick perfect square so it will be easier to simplify when you substitute. Step 4 Substitute the coordinates of the point into the inequality and simplify Step 5 Check whether the simplified inequality is true or false. If the simplified inequality is true, then shade the region that contains the point you tested. If it is false, then shade the other region

x Substitute and Simplify Into the Inequality y

 Part of Solution Set?
       
       
       
       

Note: Inequalities have one, an infinite number, or no solutions.

Example: y < √x

Step 1: Since the inequality symbol is <, the curve should be dotted.

Step 2: Graph function—Red graph

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Step 3: Test point (1, 2) Remember to try to pick a point where x is a perfect square number.

Step 4: Substitute and simplify.

y < √x
2 < √1
2 < 1

graphing calculator screen showing the graph of y = square root of x with curve as a dotted line and shading below the curve.

Step 5: This is false, so shade the other side of the graph.

x Substitute and Simplify Into the Inequality y Part of Solution Set?
1 y < √1 .5 Yes
2 y < √2 1 Yes
4 y < √4 1 Yes
9 y < √9 2 Yes

Example: y ≤ y < √x

Step 1: Since the inequality symbol is ≤, the curve should be solid.

Step 2: Graph—Red graph

Step 3: Test point, (1, 2) Remember to try to pick a point where x is a perfect square number.

Step 4: Substitute and simplify.

y ≤ √x
2 ≤ √1
2 ≤ 1

graphing calculator screen showing the graph of y = square root of x with curve as a solid line with shading below the curve.

Step 5: This is false, so shade the other side of the graph.

x Substitute and simplify into the inequality y Part of solution set?
1 y ≤ √1 .5 Yes
2 y ≤ √1 1 Yes
4 y ≤ √1 1 Yes
9 y ≤ √1 2 Yes

Below are graphs of the same function with different inequality signs.

graphing calculator screen showing the graph of y = square root of x with curve as a dotted line and shading above the curve and graphing calculator screen showing the graph of y = square root of x with curve as a solid line and shading above the curve.

Interactive popup. Assistance may be required.

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y > √x
Since the inequality symbol is <, the curve should be dotted.
Pick a test point, say (1, 2)
Substitute and simplify.
y > √x
2 > √1
2 > 1
This is true, so shade the region of the graph that contains (1,2).

x y
1 4
2 3
4 3
9 5
  • It is easiest to pick perfect square numbers for x, but you don't have to.
  • Notice that all of the y-values are greater than √x.
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Interactive popup. Assistance may be required.

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y ≥ x
Since the inequality symbol is <, the curve should be solid.
Pick a test point, say (1, 2)
Substitute and simplify.
y ≥ x
2 ≥ √1
2 ≥ 1
This is true, so shade the region of the graph that contains (1,2).

x y
0 0
1 1
2 √2
4 2
  • y can now be equal to √x, so now you can also use points on the curve.
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Is there any relationship between the symbol and shaded region of the graph?

Move your mouse over the blank to reveal the answers. (above or below)

The inequality < was shaded Interactive button. Assistance may be required. _______ below the curve.

The inequality ≤ was shaded Interactive button. Assistance may be required. _______ below the curve.

The inequality > was shaded Interactive button. Assistance may be required. _______ above the curve.

The inequality ≥ was shaded Interactive button. Assistance may be required. _______ above the curve.

Interactive popup. Assistance may be required.

What is the relationship?

For < and ≤, the shading is below the graph.
For > and ≥, the shading is above the graph. Close Pop Up

Example 1: Graph y ≤ √2x + 4

Step 1: The inequality is ≤, the curve is solid.

Step 2: Graph

graphing calculator screen showing the graph of y = square root of (2x + 4) with curve as a solid line.

Notice that the graph is only in the 1st and 2nd quadrants because y cannot be negative.

Step 3: Pick a test point: (0,3)

Step 4: Substitute and simplify.

y ≤ √2x + 4
2 ≤ √2(0) + 4
2 ≤ 1

Step 5: Since 3 ≤ 2, this is false, so shade the other side of the graph.

graphing calculator screen showing the graph of y = square root of (2x + 4) with curve as a solid line with shading below the curve.

OR

Notice the inequality is “less than” so shade below the graph.

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