3.6 Absolute Value Functions

ⓑ Solve for x: |2x–5|=3

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a…

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically reflected, and vertically shifted up 3 units.

For the function $f \left(x\right) = \left|2 x - 1\right| - 3 ,$ find the values of $x$ such that $f \left(x\right) = 0.$

How do you solve an absolute value equation?

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

Describe all numbers $x$ that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.

Describe the situation in which the distance that point $x$ is from 10 is at least 15 units. Express this set of numbers using absolute value notation.

Find the x- and y-intercepts of the graphs of each function.

$f \left(x\right) = 4 \left|x - 3\right| + 4$

Find the x- and y-intercepts of the graphs of each function.

$f \left(x\right) = - 2 \left|x + 1\right| + 6$

Find the x- and y-intercepts of the graphs of each function.

$f \left(x\right) = 2 \left|x - 1\right| - 6$

Find the x- and y-intercepts of the graphs of each function.

$f \left(x\right) = - \left|x - 9\right| + 16$

Solve the inequality.

If possible, find all values of $a$ such that there are no $y$ -intercepts for $f \left(x\right) = 2 \left|x + 1\right| + a .$

The true proportion $p$ of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using $x$ as the diameter of the bearing, write this statement using absolute value notation.

Describe all values $x$ within a distance of 3 from the number 2.

Solve $1 = 4 \left|x - 2\right| + 2.$

Find where the graph of the function $f \left(x\right) = - \left|x + 2\right| + 3$ intersects the horizontal and vertical axes.

Solve $\left|x + 2\right| \leq 6.$

Solve $- 2 \left|k - 4\right| \leq - 6.$

How do you solve an absolute value inequality algebraically?

Describe all numbers $x$ that are at a distance of $\frac{1}{2}$ from the number −4. Express this using absolute value notation.

Find all function values $f \left(x\right)$ such that the distance from $f \left(x\right)$ to the value 8 is less than 0.03 units. Express this using absolute value notation.

Solve the equations below and express the answer using set notation.

$\left|6 - x\right| = 5$

Solve the equations below and express the answer using set notation.

$\left|4 x - 2\right| = 11$

Solve the equations below and express the answer using set notation.

$3 \left|5 - x\right| = 5$

Solve the equations below and express the answer using set notation.

$5 \left|x - 4\right| - 7 = 2$

Solve the equations below and express the answer using set notation.

$2 \left|x - 3\right| + 1 = 2$

Solve the equations below and express the answer using set notation.

$\left|3 x - 2\right| = - 7$

Solve the equations below and express the answer using set notation.

$\left|\frac{1}{3} x + 5\right| = 14$

Find the x- and y-intercepts of the graphs of each function.

$f \left(x\right) = 2 \left|x + 1\right| - 10$

Find the x- and y-intercepts of the graphs of each function.

$f \left(x\right) = - 3 \left|x - 2\right| - 1$

Solve each inequality and write the solution in interval notation.

$\left|x - 2\right| > 10$

Solve each inequality and write the solution in interval notation.

$\left|3 x - 4\right| \leq 8$

Solve each inequality and write the solution in interval notation.

$\left|3 x - 5\right| \geq 13$

Solve each inequality and write the solution in interval notation.

$\left|\frac{3}{4} x - 5\right| \geq 7$

Solve the inequality.

$\left|- 2 x - \frac{2}{3} \left(x + 1\right)\right| + 3 > −1$