Solving Radical Inequalities

Worked solutions · Section 4.4, page 223 · Problems 37-52

Solving a radical inequality

\(n\) = index (the small number on the radical)

1. Isolate the radical

\(a\sqrt[n]{u} + c \le d \ \Rightarrow\ \sqrt[n]{u} \le \dfrac{d - c}{a}\)
Radical alone on one side before any power

2. Raise each side to the index

\(\left(\sqrt[n]{u}\right)^n = u\)
Same direction for a positive multiplier. Index 2: square. Index 3: cube

3. Negative multiplier flips the sign

\(-2v < 12 \ \Rightarrow\ v > -6\)
Dividing or multiplying by a negative reverses the inequality

4. Radicand of an even root

\(u \ge 0 \quad (n \text{ even})\)
No negative radicand under an even radical. Both conditions hold together

5. Odd index takes any radicand

\(\sqrt[3]{u} \ \text{defined for every} \ u\)
A cube root sets no second condition

6. Write the solution as a range

\(0 \le x < \dfrac{32}{49}\)
Lower bound from the radicand, upper bound from the inequality

Nonlinear systems

7. Substitution

\(y = x - 3 \ \Rightarrow\ \sqrt{x-3} = x - 3\)
One equation gives y; put it into the other

8. Zero-product property

\(ab = 0 \ \Rightarrow\ a = 0 \ \text{or}\ b = 0\)
Standard form first, then factor

9. Circle

\(x^2 + y^2 = r^2\)
Center at the origin, radius \(r\)

10. Check each point in both equations

\((2, 0):\ 2^2 + 0^2 = 4 \ \text{and}\ 0 = 2 - 2\)
A point must satisfy both equations

Problems 37-44: Solve the inequality.

Even index: keep the radicand nonnegative as a second condition

Problem 37

\(2\sqrt[3]{x} - 5 \ge 3\)

Rules used:
  • 1. Isolate the radical
  • 2. Raise each side to the index
  • 5. Odd index takes any radicand
Step 1. Isolate the radical.
\(2\sqrt[3]{x} \ge 8\)
\(\sqrt[3]{x} \ge 4\)
Step 2. Cube each side.
\(x \ge 4^3\)
Step 3. Odd index, so no second condition.
A cube root is defined for every \(x\)
Solution
\(x \ge 64\)
Problem 38

\(\sqrt[3]{x-4} \le 5\)

Rules used:
  • 2. Raise each side to the index
  • 5. Odd index takes any radicand
Step 1. Cube each side.
\(x - 4 \le 5^3\)
\(x - 4 \le 125\)
Step 2. Add 4 to each side.
\(x \le 129\)
Solution
\(x \le 129\)
Problem 39

\(4\sqrt{x-2} > 20\)

Rules used:
  • 1. Isolate the radical
  • 2. Raise each side to the index
  • 4. Radicand of an even root
  • 6. Write the solution as a range
Step 1. Isolate the radical.
\(\sqrt{x-2} > 5\)
Step 2. Square each side.
\(x - 2 > 25\)
\(x > 27\)
Step 3. Check the radicand.
\(x - 2 \ge 0\)
\(x \ge 2\), already covered by \(x > 27\)
Solution
\(x > 27\)
Problem 40

\(7\sqrt{x+1} < 9\)

Rules used:
  • 1. Isolate the radical
  • 2. Raise each side to the index
  • 4. Radicand of an even root
  • 6. Write the solution as a range
Step 1. Isolate the radical.
\(\sqrt{x+1} < \dfrac{9}{7}\)
Step 2. Square each side.
\(x + 1 < \dfrac{81}{49}\)
\(x < \dfrac{32}{49}\)
Step 3. Check the radicand.
\(x + 1 \ge 0\)
\(x \ge -1\), which sets the lower bound
Solution
\(-1 \le x < \dfrac{32}{49}\)
Problem 41

\(2\sqrt{x} + 3 \le 8\)

Rules used:
  • 1. Isolate the radical
  • 2. Raise each side to the index
  • 4. Radicand of an even root
  • 6. Write the solution as a range
Step 1. Isolate the radical.
\(2\sqrt{x} \le 5\)
\(\sqrt{x} \le \dfrac{5}{2}\)
Step 2. Square each side.
\(x \le \dfrac{25}{4}\)
Step 3. Check the radicand.
\(x \ge 0\)
The lower bound
Solution
\(0 \le x \le \dfrac{25}{4}\)
Problem 42

\(\sqrt[3]{x+7} \ge 3\)

Rules used:
  • 2. Raise each side to the index
  • 5. Odd index takes any radicand
Step 1. Cube each side.
\(x + 7 \ge 27\)
Step 2. Subtract 7 from each side.
\(x \ge 20\)
Solution
\(x \ge 20\)
Problem 43

\(-2\sqrt[3]{x+4} < 12\)

Rules used:
  • 1. Isolate the radical
  • 3. Negative multiplier flips the sign
  • 2. Raise each side to the index
  • 5. Odd index takes any radicand
Step 1. Divide each side by -2 and flip the sign.
\(\sqrt[3]{x+4} > -6\)
Step 2. Cube each side.
\(x + 4 > -216\)
Step 3. Subtract 4 from each side.
\(x > -220\)
Solution
\(x > -220\)
Problem 44

\(-0.25\sqrt{x-6} \le -3\)

Rules used:
  • 1. Isolate the radical
  • 3. Negative multiplier flips the sign
  • 2. Raise each side to the index
  • 4. Radicand of an even root
Step 1. Divide each side by -0.25 and flip the sign.
\(\sqrt{x-6} \ge 12\)
Step 2. Square each side.
\(x - 6 \ge 144\)
\(x \ge 150\)
Step 3. Check the radicand.
\(x - 6 \ge 0\)
\(x \ge 6\), already covered
Solution
\(x \ge 150\)

Problems 45-46: Modeling with mathematics

Problem 45

\(\ell = 54d^{3/2}\)

Rules used:
  • 1. Isolate the radical
  • 2. Raise each side to the index
Step 1. Substitute 3 for the length.
\(3 = 54d^{3/2}\)
Step 2. Divide each side by 54.
\(d^{3/2} = \dfrac{1}{18}\)
Step 3. Raise each side to the two-thirds.
\(d = \left(\dfrac{1}{18}\right)^{2/3} \approx 0.15\)
Answer
\(d \approx 0.15\)
About 0.15 inch across
Problem 46

\(t = 0.5\sqrt{h}\)

Rules used:
  • 1. Isolate the radical
  • 2. Raise each side to the index
Step 1. Solve the model for the height.
\(\sqrt{h} = 2t\)
\(h = 4t^2\)
Step 2. Snowboarder, t = 1.21.
\(h = 4(1.21)^2 \approx 5.86\)
Step 3. Kangaroo, t = 0.81.
\(h = 4(0.81)^2 \approx 2.62\)
Step 4. Double each hang time.
\(h = 4(2.42)^2 \approx 23.43\)
\(h = 4(1.62)^2 \approx 10.50\)
Step 5. Compare.
Doubling \(t\) multiplies \(h\) by 4, since \(h = 4t^2\)
Answer
\(h = 4t^2\)
Snowboarder about 5.86 feet, kangaroo about 2.62 feet

Problems 47-52: Solve the nonlinear system. Justify your answer with a graph.

Problem 47

\(y = \sqrt{x-3},\ y = x - 3\)

Rules used:
  • 7. Substitution
  • 8. Zero-product property
  • 10. Check each point in both equations
Step 1. Substitute for y.
\(\sqrt{x-3} = x - 3\)
Step 2. Square each side.
\(x - 3 = (x-3)^2\)
Step 3. Write in standard form and factor.
\(0 = (x-3)(x-4)\)
Step 4. Find each y.
\(x = 3\)
\(x = 4\)
\(y = 0\) and \(y = 1\)
Solutions
\((3, 0) \ \text{and}\ (4, 1)\)
Problem 48

\(y = \sqrt{4x+17},\ y = x + 5\)

Rules used:
  • 7. Substitution
  • 8. Zero-product property
  • 10. Check each point in both equations
Step 1. Substitute for y and square each side.
\(4x + 17 = (x+5)^2\)
Step 2. Write in standard form and factor.
\(0 = x^2 + 6x + 8\)
\(0 = (x+2)(x+4)\)
Step 3. Find each y.
\(x = -2\)
\(x = -4\)
\(y = 3\) and \(y = 1\)
Solutions
\((-2, 3) \ \text{and}\ (-4, 1)\)
Problem 49

\(x^2 + y^2 = 4,\ y = x - 2\)

Rules used:
  • 9. Circle
  • 7. Substitution
  • 8. Zero-product property
  • 10. Check each point in both equations
Step 1. Substitute for y.
\(x^2 + (x-2)^2 = 4\)
Step 2. Expand and simplify.
\(2x^2 - 4x = 0\)
\((2x)(x - 2) = 0\)
Step 3. Find each y.
\(x = 0\)
\(x = 2\)
\(y = -2\) and \(y = 0\)
Solutions
\((0, -2) \ \text{and}\ (2, 0)\)
Problem 50

\(x^2 + y^2 = 25,\ y = -\dfrac{3}{4}x + \dfrac{25}{4}\)

Rules used:
  • 9. Circle
  • 7. Substitution
  • 8. Zero-product property
  • 10. Check each point in both equations
Step 1. Substitute for y.
\(x^2 + \left(-\dfrac{3}{4}x + \dfrac{25}{4}\right)^2 = 25\)
Step 2. Clear the fractions and simplify.
\(25x^2 - 150x + 225 = 0\)
\(x^2 - 6x + 9 = 0\)
Step 3. Factor.
\((x-3)^2 = 0\)
\(x = 3\)
Step 4. Find y.
\(y = 4\)
One point: the line touches the circle
Solution
\((3, 4)\)
Problem 51

\(x^2 + y^2 = 1,\ y = \dfrac{1}{2}x^2 - 1\)

Rules used:
  • 9. Circle
  • 7. Substitution
  • 8. Zero-product property
  • 10. Check each point in both equations
Step 1. Solve the second equation for x squared.
\(x^2 = 2y + 2\)
Step 2. Substitute into the circle.
\(2y + 2 + y^2 = 1\)
\(y^2 + 2y + 1 = 0\)
Step 3. Factor.
\((y+1)^2 = 0\)
\(y = -1\)
Step 4. Find x.
\(x^2 = 0\)
\(x = 0\)
Solution
\((0, -1)\)
Problem 52

\(x^2 + y^2 = 4,\ y^2 = x + 2\)

Rules used:
  • 9. Circle
  • 7. Substitution
  • 8. Zero-product property
  • 10. Check each point in both equations
Step 1. Substitute for y squared.
\(x^2 + x + 2 = 4\)
Step 2. Write in standard form and factor.
\(x^2 + x - 2 = 0\)
\((x+2)(x-1) = 0\)
Step 3. Find each y.
\(x = -2\)
\(x = 1\)
\(y = 0\); and \(y^2 = 3\), so \(y = \pm\sqrt{3}\)
Solutions
\((-2, 0),\ (1, \sqrt{3}),\ (1, -\sqrt{3})\)

Quick checklist

  1. Radical alone on one side? Isolate it before any power
  2. Multiplied or divided by a negative? Flip the inequality sign
  3. Even index? Radicand \(\ge 0\) is a second condition
  4. Odd index? No second condition
  5. Two conditions? Write the answer as one range
  6. Nonlinear system? Substitute, factor, then find every \(y\)
  7. Each point checked in both equations? The graph shows how many to expect