Algebra Examples

$3 + 4 | 3 x + 7 | \geq - 89$

Step 1

Write $3 + 4 | 3 x + 7 | \geq - 89$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$3 x + 7 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $7$ from both sides of the inequality.

$3 x \geq - 7$

Step 1.2.2

Divide each term in $3 x \geq - 7$ by $3$ and simplify.

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Step 1.2.2.1

Divide each term in $3 x \geq - 7$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 7}{3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 7}{3}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 7}{3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{7}{3}$

Step 1.3

In the piece where $3 x + 7$ is non-negative, remove the absolute value.

$3 + 4 (3 x + 7) \geq - 89$

Step 1.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$3 x + 7 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $7$ from both sides of the inequality.

$3 x < - 7$

Step 1.5.2

Divide each term in $3 x < - 7$ by $3$ and simplify.

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Step 1.5.2.1

Divide each term in $3 x < - 7$ by $3$ .

$\frac{3 x}{3} < \frac{- 7}{3}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 7}{3}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 7}{3}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Move the negative in front of the fraction.

$x < - \frac{7}{3}$

Step 1.6

In the piece where $3 x + 7$ is negative, remove the absolute value and multiply by $- 1$ .

$3 + 4 (- (3 x + 7)) \geq - 89$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 + 4 (3 x + 7) \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.8

Simplify $3 + 4 (3 x + 7) \geq - 89$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 3 + 4 (3 x) + 4 \cdot 7 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.8.1.2

Multiply $3$ by $4$ .

${\begin{cases} 3 + 12 x + 4 \cdot 7 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.8.1.3

Multiply $4$ by $7$ .

${\begin{cases} 3 + 12 x + 28 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.8.2

Add $3$ and $28$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x + 7)) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.9

Simplify $3 + 4 (- (3 x + 7)) \geq - 89$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- (3 x) - 1 \cdot 7) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.9.1.2

Multiply $3$ by $- 1$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- 3 x - 1 \cdot 7) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $7$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- 3 x - 7) \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.9.1.4

Apply the distributive property.

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 + 4 (- 3 x) + 4 \cdot - 7 \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.9.1.5

Multiply $- 3$ by $4$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 - 12 x + 4 \cdot - 7 \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.9.1.6

Multiply $4$ by $- 7$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ 3 - 12 x - 28 \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 1.9.2

Subtract $28$ from $3$ .

${\begin{cases} 12 x + 31 \geq - 89 & x \geq - \frac{7}{3} \\ - 12 x - 25 \geq - 89 & x < - \frac{7}{3} \end{cases}$

Step 2

Solve $12 x + 31 \geq - 89$ when $x \geq - \frac{7}{3}$ .

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Step 2.1

Solve $12 x + 31 \geq - 89$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $31$ from both sides of the inequality.

$12 x \geq - 89 - 31$

Step 2.1.1.2

Subtract $31$ from $- 89$ .

$12 x \geq - 120$

Step 2.1.2

Divide each term in $12 x \geq - 120$ by $12$ and simplify.

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Step 2.1.2.1

Divide each term in $12 x \geq - 120$ by $12$ .

$\frac{12 x}{12} \geq \frac{- 120}{12}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $12$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} \geq \frac{- 120}{12}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 120}{12}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $- 120$ by $12$ .

$x \geq - 10$

Step 2.2

Find the intersection of $x \geq - 10$ and $x \geq - \frac{7}{3}$ .

$x \geq - \frac{7}{3}$

Step 3

Solve $- 12 x - 25 \geq - 89$ when $x < - \frac{7}{3}$ .

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Step 3.1

Solve $- 12 x - 25 \geq - 89$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $25$ to both sides of the inequality.

$- 12 x \geq - 89 + 25$

Step 3.1.1.2

Add $- 89$ and $25$ .

$- 12 x \geq - 64$

Step 3.1.2

Divide each term in $- 12 x \geq - 64$ by $- 12$ and simplify.

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Step 3.1.2.1

Divide each term in $- 12 x \geq - 64$ by $- 12$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 12 x}{- 12} \leq \frac{- 64}{- 12}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 12$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 12 x}{- 12} \leq \frac{- 64}{- 12}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 64}{- 12}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Cancel the common factor of $- 64$ and $- 12$ .

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Step 3.1.2.3.1.1

Factor $- 4$ out of $- 64$ .

$x \leq \frac{- 4 (16)}{- 12}$

Step 3.1.2.3.1.2

Cancel the common factors.

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Step 3.1.2.3.1.2.1

Factor $- 4$ out of $- 12$ .

$x \leq \frac{- 4 \cdot 16}{- 4 \cdot 3}$

Step 3.1.2.3.1.2.2

Cancel the common factor.

$x \leq \frac{- 4 \cdot 16}{- 4 \cdot 3}$

Step 3.1.2.3.1.2.3

Rewrite the expression.

$x \leq \frac{16}{3}$

Step 3.2

Find the intersection of $x \leq \frac{16}{3}$ and $x < - \frac{7}{3}$ .

$x < - \frac{7}{3}$

Step 4

Find the union of the solutions.

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 6