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A.9(E)·Algebra II·Square Root FunctionsSupporting

Square Root Inequalities

Determine solutions of square root inequalities using tables and graphs.

Students test ordered pairs and tables against square-root inequalities, respecting the radical domain and distinguishing boundary values from the solution region.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Algebra II released practice50 minutes

Learn the concept

Check the domain first

For y≤a−x+ky\le\sqrt{a-x}+k, require a−x≥0a-x\ge0. An input outside the real domain cannot produce a solution pair, regardless of its y-value.

Compute and compare

Substitute only the x-coordinate to find the boundary height. Then compare the pair's y-coordinate with that height. For ≤\le, points at or below the boundary pass.

Read the whole table and graph

Test every row. On a graph, use a solid boundary for ≤ or ≥ and a dashed boundary for < or >. Shade only where the square-root expression is defined.

Worked example

Do all pairs in the table (0,10)(0,10), (9,9)(9,9), (21,5)(21,5) satisfy y≤25−x+5y\le\sqrt{25-x}+5?

  1. All three inputs are at most 25, so each radicand is nonnegative.
  2. At x=0x=0, the boundary is 25+5=10\sqrt{25}+5=10, and 10≤1010\le10.
  3. At x=9x=9, the boundary is 16+5=9\sqrt{16}+5=9, and 9≤99\le9.
  4. At x=21x=21, the boundary is 4+5=7\sqrt4+5=7, and 5≤75\le7.

Yes. Every row satisfies the inequality.

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Identifying Parent Functions

UT IPSI · Kid2Kid / Texas Education Agency

Recognize the square-root parent graph before interpreting its boundary and solution region.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. Which pairs satisfy y≤20−x+3y\le\sqrt{20-x}+3: (4,7)(4,7), (11,7)(11,7), and (19,0)(19,0)?

    Show answer and reasoning

    (4,7)(4,7) and (19,0)(19,0)

    The boundary heights are 7, 6, and 4. The middle pair fails because 7 is greater than 6.

  2. 2. Can (13,−100)(13,-100) satisfy y<12−x+4y<\sqrt{12-x}+4 over the reals?

    Show answer and reasoning

    No.

    The radicand is −1, so the boundary expression is not real. A very small y-value cannot repair an invalid input.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 38 · 1 point

Which table contains only values in the solution set for y≤25−x+11y \le \sqrt{25 - x} + 11?

  1. Table F: (−24, 19), (−11, 17), (9, 10), (24, 9)
  2. Table G: (−24, 18), (−11, 16), (9, 11), (24, 9)
  3. Table H: (−24, 19), (−11, 18), (9, −18), (24, −19)
  4. Table J: (−24, 9), (−11, 10), (9, 15), (24, 17)
Show STAAR answer and explanation

Table G: (−24, 18), (−11, 16), (9, 11), (24, 9)

The boundary values are 49+11=18\sqrt{49}+11=18 at x=−24x=-24, 36+11=17\sqrt{36}+11=17 at x=−11x=-11, 16+11=15\sqrt{16}+11=15 at x=9x=9, and 1+11=12\sqrt{1}+11=12 at x=24x=24. Every y-value in Table G (18, 16, 11, 9) is at or below its boundary value.

Table F: (−24, 19), (−11, 17), (9, 10), (24, 9): (−24,19)(-24,19) fails because 19>1819>18.

Table H: (−24, 19), (−11, 18), (9, −18), (24, −19): (−24,19)(-24,19) and (−11,18)(-11,18) are above the boundary values 18 and 17.

Table J: (−24, 9), (−11, 10), (9, 15), (24, 17): (24,17)(24,17) fails because 17>1217>12.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Check that each input produces a real square root.
  • Test every row of a table in a square-root inequality.
  • Interpret included boundaries and shading on a graph.

Instructional sequence

  1. 1

    Engage

    5 min

    Ask whether a point below a curve must lie on the curve to satisfy a ≤ inequality.

  2. 2

    Direct instruction

    12 min

    Model the domain check, boundary calculation, and output comparison.

  3. 3

    Guided practice

    13 min

    Evaluate candidate tables and reject any table containing a failing row.

  4. 4

    Graph connection

    13 min

    Match algebraic checks to points on, above, and below the boundary.

  5. 5

    Closure

    7 min

    Students justify one accepted and one rejected point.

Misconceptions to address

““Only points on the square-root curve are solutions.””

Teacher move: The equation describes the boundary; an inequality can include an entire region above or below it.

““One passing row proves the whole table works.””

Teacher move: A table contains only solutions only when every row passes.