Graph f(x)<|4x+1|-3

$f (x) < | 4 x + 1 | - 3$

Step 1

Subtract $f (x)$ from both sides of the inequality.

$0 < | 4 x + 1 | - 3 - f (x)$

Step 2

Find the slope and the y-intercept for the boundary line.

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Rewrite so $x$ is on the left side of the inequality.

$| 4 x + 1 | - 3 - f (x) > 0$

Write $| 4 x + 1 | - 3 - f (x) > 0$ as a piecewise.

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

$4 x + 1 \geq 0$

Solve the inequality.

Subtract $1$ from both sides of the inequality.

$4 x \geq - 1$

Divide each term in $4 x \geq - 1$ by $4$ and simplify.

Divide each term in $4 x \geq - 1$ by $4$ .

$\frac{4 x}{4} \geq \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{- 1}{4}$

Divide $x$ by $1$ .

$x \geq \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{1}{4}$

In the piece where $4 x + 1$ is non-negative, remove the absolute value.

$4 x + 1 - 3 - f (x) > 0$

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

$4 x + 1 < 0$

Solve the inequality.

Subtract $1$ from both sides of the inequality.

$4 x < - 1$

Divide each term in $4 x < - 1$ by $4$ and simplify.

Divide each term in $4 x < - 1$ by $4$ .

$\frac{4 x}{4} < \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} < \frac{- 1}{4}$

Divide $x$ by $1$ .

$x < \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x < - \frac{1}{4}$

In the piece where $4 x + 1$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x + 1) - 3 - f (x) > 0$

Write as a piecewise.

${\begin{cases} 4 x + 1 - 3 - f (x) > 0 & x \geq - \frac{1}{4} \\ - (4 x + 1) - 3 - f (x) > 0 & x < - \frac{1}{4} \end{cases}$

Subtract $3$ from $1$ .

${\begin{cases} 4 x - 2 - f (x) > 0 & x \geq - \frac{1}{4} \\ - (4 x + 1) - 3 - f (x) > 0 & x < - \frac{1}{4} \end{cases}$

Simplify $- (4 x + 1) - 3 - f (x) > 0$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 x - 2 - f (x) > 0 & x \geq - \frac{1}{4} \\ - (4 x) - 1 \cdot 1 - 3 - f (x) > 0 & x < - \frac{1}{4} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x - 2 - f (x) > 0 & x \geq - \frac{1}{4} \\ - 4 x - 1 \cdot 1 - 3 - f (x) > 0 & x < - \frac{1}{4} \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} 4 x - 2 - f (x) > 0 & x \geq - \frac{1}{4} \\ - 4 x - 1 - 3 - f (x) > 0 & x < - \frac{1}{4} \end{cases}$

Subtract $3$ from $- 1$ .

${\begin{cases} 4 x - 2 - f (x) > 0 & x \geq - \frac{1}{4} \\ - 4 x - 4 - f (x) > 0 & x < - \frac{1}{4} \end{cases}$

Solve $4 x - 2 - f (x) > 0$ when $x \geq - \frac{1}{4}$ .

Solve $4 x - 2 - f (x) > 0$ for $x$ .

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

Add $2$ to both sides of the inequality.

$4 x - f (x) > 2$

Add $f (x)$ to both sides of the inequality.

$4 x > 2 + f (x)$

Divide each term in $4 x > 2 + f (x)$ by $4$ and simplify.

Divide each term in $4 x > 2 + f (x)$ by $4$ .

$\frac{4 x}{4} > \frac{2}{4} + \frac{f (x)}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} > \frac{2}{4} + \frac{f (x)}{4}$

Divide $x$ by $1$ .

$x > \frac{2}{4} + \frac{f (x)}{4}$

Simplify the right side.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$x > \frac{2 (1)}{4} + \frac{f (x)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x > \frac{2 \cdot 1}{2 \cdot 2} + \frac{f (x)}{4}$

Cancel the common factor.

$x > \frac{2 \cdot 1}{2 \cdot 2} + \frac{f (x)}{4}$

Rewrite the expression.

$x > \frac{1}{2} + \frac{f (x)}{4}$

Find the intersection of $x > \frac{1}{2} + \frac{f (x)}{4}$ and $x \geq - \frac{1}{4}$ .

$x > \frac{1}{2} + \frac{f (x)}{4}$ and $x \geq - \frac{1}{4}$

Solve $- 4 x - 4 - f (x) > 0$ when $x < - \frac{1}{4}$ .

Solve $- 4 x - 4 - f (x) > 0$ for $x$ .

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

Add $4$ to both sides of the inequality.

$- 4 x - f (x) > 4$

Add $f (x)$ to both sides of the inequality.

$- 4 x > 4 + f (x)$

Divide each term in $- 4 x > 4 + f (x)$ by $- 4$ and simplify.

Divide each term in $- 4 x > 4 + f (x)$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} < \frac{4}{- 4} + \frac{f (x)}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} < \frac{4}{- 4} + \frac{f (x)}{- 4}$

Divide $x$ by $1$ .

$x < \frac{4}{- 4} + \frac{f (x)}{- 4}$

Simplify the right side.

Simplify each term.

Divide $4$ by $- 4$ .

$x < - 1 + \frac{f (x)}{- 4}$

Move the negative in front of the fraction.

$x < - 1 - \frac{f (x)}{4}$

Find the intersection of $x < - 1 - \frac{f (x)}{4}$ and $x < - \frac{1}{4}$ .

$x < N o (Min i m u m)$

Find the union of the solutions.

$x > N o Max i m u m$ or $x < N o Min i m u m$

The equation is not linear, so a constant slope does not exist.

Not Linear

Step 3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $| 4 x + 1 | - 3 - f (x)$ .

$0 < | 4 x + 1 | - 3 - f (x)$

Step 4

Graph f(x)<|4x+1|-3

Related Questions

Support LCMGCF.com by disabling your adblocker.