Solve for x |2x+4|>=|x+5|

$| 2 x + 4 | \geq | x + 5 |$

Step 1

Replace $\geq$ with $=$ in $| 2 x + 4 | \geq | x + 5 |$ .

$| 2 x + 4 | = | x + 5 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$2 x + 4 = x + 5$

$2 x + 4 = - (x + 5)$

$- (2 x + 4) = x + 5$

$- (2 x + 4) = - (x + 5)$

Step 3

After simplifying, there are only two unique equations to be solved.

$2 x + 4 = x + 5$

$2 x + 4 = - (x + 5)$

Step 4

Solve $2 x + 4 = x + 5$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$2 x + 4 - x = 5$

Subtract $x$ from $2 x$ .

$x + 4 = 5$

Move all terms not containing $x$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$x = 5 - 4$

Subtract $4$ from $5$ .

$x = 1$

Step 5

Solve $2 x + 4 = - (x + 5)$ for $x$ .

Simplify $- (x + 5)$ .

Rewrite.

$2 x + 4 = 0 + 0 - (x + 5)$

Simplify by adding zeros.

$2 x + 4 = - (x + 5)$

Apply the distributive property.

$2 x + 4 = - x - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$2 x + 4 = - x - 5$

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$2 x + 4 + x = - 5$

Add $2 x$ and $x$ .

$3 x + 4 = - 5$

Move all terms not containing $x$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$3 x = - 5 - 4$

Subtract $4$ from $- 5$ .

$3 x = - 9$

Divide each term in $3 x = - 9$ by $3$ and simplify.

Divide each term in $3 x = - 9$ by $3$ .

$\frac{3 x}{3} = \frac{- 9}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 9}{3}$

Divide $x$ by $1$ .

$x = \frac{- 9}{3}$

Simplify the right side.

Divide $- 9$ by $3$ .

$x = - 3$

Step 6

List all of the solutions.

$x = 1, - 3$

Step 7

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 1$

$x > 1$

Step 8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 3$ to see if it makes the inequality true.

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

$| 2 (- 6) + 4 | \geq | (- 6) + 5 |$

The left side $8$ is greater than the right side $1$ , which means that the given statement is always true.

True

Test a value on the interval $- 3 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$| 2 (0) + 4 | \geq | (0) + 5 |$

The left side $4$ is less than the right side $5$ , which means that the given statement is false.

False

Test a value on the interval $x > 1$ to see if it makes the inequality true.

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$| 2 (4) + 4 | \geq | (4) + 5 |$

The left side $12$ is greater than the right side $9$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 3$ True

$- 3 < x < 1$ False

$x > 1$ True

$x < - 3$ True

$- 3 < x < 1$ False

$x > 1$ True

Step 9

The solution consists of all of the true intervals.

$x \leq - 3$ or $x \geq 1$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 3 or x \geq 1$

Interval Notation:

$(- \infty, - 3] \cup [1, \infty)$

Step 11

Solve for x |2x+4|>=|x+5|

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