Understanding Radical Equations and Techniques for Solving Radical Inequalities

Variable in a Radicand	Variable With a Rational Exponent
$x - 2 = x + 2$	$x^{\frac{1}{2}} = 3 x - 5$
$3 2 x = x + 1$	$1 - 3 x = (x - 1)^{\frac{1}{3}}$
$3 = 4 2 x + 1$	$(2 x)^{\frac{1}{4}} = 4 x - 9$

Variable in a Radicand

Variable With a Rational Exponent

x - 2 = x + 2

x^{\frac{1}{2}} = 3 x - 5

3 2 x = x + 1

1 - 3 x = (x - 1)^{\frac{1}{3}}

3 = 4 2 x + 1

(2 x)^{\frac{1}{4}} = 4 x - 9

	Substitute	Simplify
$x = 1$	$2 = ? 1 + 2 (1) - 1$	$2 = 2$ $✓$
$x = 5$	$2 = ? 5 + 2 (5) - 1$	$2 \neq = 8$ $\times$

Substitute

Simplify

x = 1

2 = ? 1 + 2 (1) - 1

2 = 2

✓

x = 5

2 = ? 5 + 2 (5) - 1

2 \neq = 8

\times

	Substitute	Simplify
$x_{1} = 5$	$5 + - 5 + 14 = ? 2$	$8 \neq = 2 \times$
$x_{2} = - 2$	$- 2 + - (- 2) + 14 = ? 2$	$2 = 2 ✓$

Substitute

Simplify

x_{1} = 5

5 + - 5 + 14 = ? 2

8 \neq = 2 \times

x_{2} = - 2

- 2 + - (- 2) + 14 = ? 2

2 = 2 ✓

	$y = x + 5$	$y = x + 3$
$x$	$x + 5$	$y$	$x + 3$	$y$
$- 5$	$- 5 + 5$	$0$	$- 5 + 3$	$- 2$
$- 4$	$- 4 + 5$	$1$	$- 4 + 3$	$- 1$
$- 1$	$- 1 + 5$	$2$	$- 1 + 3$	$2$
$4$	$4 + 5$	$3$	$4 + 3$	$7$

y = x + 5

y = x + 3

x

x + 5

y

x + 3

y

- 5

- 5 + 5

0

- 5 + 3

- 2

- 4

- 4 + 5

1

- 4 + 3

- 1

- 1

- 1 + 5

2

- 1 + 3

2

4

4 + 5

3

4 + 3

7

	$y = 2 x - 6$	$y = 1 + x - 2$
$x$	$2 x - 6$	$y$	$1 + x - 2$	$y$
$2$	-	-	$1 + 2 - 2$	$1$
$3$	$2 (3) - 6$	$0$	$1 + 3 - 2$	$2$
$5$	$2 (5) - 6$	$2$	$1 + 5 - 2$	$\approx 2.73$
$6$	$2 (6) - 6$	$\approx 2.45$	$1 + 6 - 2$	$3$
$11$	$2 (11) - 6$	$4$	$1 + 11 - 2$	$4$

y = 2 x - 6

y = 1 + x - 2

x

2 x - 6

y

1 + x - 2

y

2

1 + 2 - 2

1

3

2 (3) - 6

0

1 + 3 - 2

2

5

2 (5) - 6

2

1 + 5 - 2

\approx 2.73

6

2 (6) - 6

\approx 2.45

1 + 6 - 2

3

11

2 (11) - 6

4

1 + 11 - 2

4

Variable in a Radicand	Variable With a Rational Exponent
$x - 3 < x + 3$	$x^{\frac{1}{2}} > x - 7$
$3 x > 2 x + 1$	$1 - x \leq (x + 1)^{\frac{1}{3}}$
$3 \geq 4 4 x + 1$	$(3 x)^{\frac{1}{4}} < 9 x - 4$

Variable in a Radicand

Variable With a Rational Exponent

x - 3 < x + 3

x^{\frac{1}{2}} > x - 7

3 x > 2 x + 1

1 - x \leq (x + 1)^{\frac{1}{3}}

3 \geq 4 4 x + 1

(3 x)^{\frac{1}{4}} < 9 x - 4

$3 x + 9 - 2 \leq 7$
$x$	Substitute	Simplify
$- 4$	$3 (- 4) + 9 - 2 \leq ? 7$	$- 3 ≰ 7 \times$
$0$	$3 (0) + 9 - 2 \leq ? 7$	$1 \leq 7 ✓$
$25$	$3 (25) + 9 - 2 \leq ? 7$	$7.2 ≰ 7 \times$

3 x + 9 - 2 \leq 7

x

Substitute

Simplify

- 4

3 (- 4) + 9 - 2 \leq ? 7

- 3 ≰ 7 \times

0

3 (0) + 9 - 2 \leq ? 7

1 \leq 7 ✓

25

3 (25) + 9 - 2 \leq ? 7

7.2 ≰ 7 \times

$2 x - 4 + 4 < 8$
$x$	Substitute	Simplify
$0$	$2 (0) - 4 + 4 < 8$	$- 4 ≮ 4 \times$
$4$	$2 (4) - 4 + 4 < 8$	$6 < 8 ✓$
$20$	$2 (20) - 4 + 4 < 8$	$10 ≮ 8 \times$

2 x - 4 + 4 < 8

x

Substitute

Simplify

0

2 (0) - 4 + 4 < 8

- 4 ≮ 4 \times

4

2 (4) - 4 + 4 < 8

6 < 8 ✓

20

2 (20) - 4 + 4 < 8

10 ≮ 8 \times

$10 t + 2 t = 10 2 t$
$t$	Substitute	Simplify
$0$	$10 0 + 2 (0) = ? 10 2 (0)$	$0 = 0 ✓$
$2$	$10 2 + 2 (2) = ? 10 2 (2)$	$20 = 20 ✓$

10 t + 2 t = 10 2 t

t

Substitute

Simplify

0

10 0 + 2 (0) = ? 10 2 (0)

0 = 0 ✓

2

10 2 + 2 (2) = ? 10 2 (2)

20 = 20 ✓

Solve each equation.

x - 5 + 7 = 12

3 + 9 z^{\frac{1}{2}} = 0

Solving a radical equation usually involves three main steps.

Isolate the radical on one side of the equation.
Raise each side of the equation to a power equal to the radical's index to eliminate the radical.
Solve the resulting equation. Remember to check the results!

Now we can analyze the given radical equation. sqrt(x-5)+7=12 First, let's isolate the expression sqrt(x-5) on one side.

The index of the radical is 2. Therefore, we will raise each side of the equation to the power of 2.

Next, we will check the solution by substituting 30 for x into the original equation. If the substitution produces a true statement, we know that our answer is correct. Otherwise, x=30 is an extraneous solution.

Because our substitution produced a true statement, we know that our answer is correct.

Consider the given equation with a rational exponent. 3+9z^(12)=0 Let's begin by isolating z^(12).

In order to remove the rational exponent, we will raise each side of the equation to the reciprocal of the exponent. This means that we will raise both sides to the power of 2.

Next, we will check the solution by substituting 19 for z into the original equation. If the substitution produces a true statement, we know that our answer is correct. Otherwise, z= 19 is an extraneous solution.

Because our substitution produced a false statement, z= 19 is an extraneous solution. Therefore, the equation has no solution.

Solve the radical equation.

3 x - 3 = 2

Solving a radical equation usually involves three main steps.

Isolate the radical on one side of the equation.
Raise each side of the equation to a power equal to the radical's index to eliminate the radical.
Solve the resulting equation. Remember to check the results!

Now we can analyze the given radical equation. sqrt(x-3)=2 We see that the radical expression is already isolated, and that the index is equal to 3. Therefore, we will raise each side of the equation to the power of 3.

Next, we will check the solution by substituting 11 for x into the original equation. If the substitution produces a true statement, we know that our answer is correct. Otherwise, x=11 is an extraneous solution.

Because our substitution produced a true statement, we know that our answer is correct.

Solve each radical equation.

2 4 y - 2 + 7 = 11

(3 x - 8)^{\frac{1}{4}} + 2 = 1

Solving a radical equation usually involves three main steps.

Isolate the radical on one side of the equation.
Raise each side of the equation to a power equal to the radical's index to eliminate the radical.
Solve the resulting equation. Remember to check the results!

Now we can analyze the given radical equation. 2sqrt(y-2)+7=11 First, let's the radical expression sqrt(y-2) on one side of the equation.

Now that the expression sqrt(y-2)=2 is isolated, we will raise each side of the equation to the power of 4.

Next, we will check the solution by substituting 18 for y into the original equation. If the substitution produces a true statement, we know that our answer is correct. Otherwise, y=18 is an extraneous solution.

Because our substitution produced a true statement, we know that our answer is correct.

This time we have an equation that involves a rational exponent. (3x-8)^(14)+2=1 Let's begin by isolating the expression with a rational exponent on one side of the equation. (3x-8)^(14)+2=1 ⇕ (3x-8)^(14)=- 1 In order to eliminate the power, we will raise each side of the equation to the power of 4.

Next, we will check the solution by substituting 3 for x into the original equation. If the substitution produces a true statement, we know that our answer is correct. Otherwise, x=3 is an extraneous solution.

Because our substitution produced a false statement, we know that x=3 is an extraneous solution.

Solve the radical equation.

t + 7 = 3 t - 11

Solving a radical equation usually involves three main steps.

Isolate the radical on one side of the equation.
Raise each side of the equation to a power equal to the radical's index to eliminate the radical.
Solve the resulting equation. Remember to check the results!

Now we can analyze the given radical equation. sqrt(t+7)=sqrt(3t-11) In this equation, each side contains a single radical expression. The index of these radicals is 2. Therefore, we will raise each side of the equation to the power of 2.

Next, we will check the solution by substituting 9 for t into the original equation. If the substitution produces a true statement, we know that our answer is correct. Otherwise, t=9 is an extraneous solution.

Because our substitution produced a true statement, we know that our answer is correct.

Solve the radical inequality.

2 x + 4 - 5 \leq 5

Write the answer as a compound inequality.

Before solving the radical inequality, recall that the radicand of a square root must be greater than or equal to 0. sqrt(2x+4)-5 ≤ 5 Let's start by identifying the values of the variable x for which the inequality is defined.

After we solve the inequality, we can combine this information with our solution to find the complete solution set of the inequality. Let's now solve the inequality. To do so, we will add 5 to both sides to isolate the radical expression. Then, we can raise both sides to the power of 2.

Now we can combine the two intervals.

The intersection of both solutions, which is the solution set of the given inequality, is -2 ≤ x ≤ 48.

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