3.06 Absolute value equations and inequalities
Write an absolute value equation for each of the following:
Representing all numbers x whose distance from 0 is 9 units.
5 x - 2 = 3 or 5 x - 2 = - 3
Determine how many solutions there are for each of the following absolute value equations:
\left|x\right| = a, where a is a positive number.
For what value(s) of b does the equation \left|x\right| = b have only one solution?
Roxanne performed the following steps to solve the equation 4 \left|x\right| = - 2:
Which, if any, of the numbers that Roxanne wrote down as solutions actually satisfy the equation 4 \left|x\right| = - 2? Explain your answer.
| 4\left|x\right| = - 2 | |
| Step 1: | 4 x = - 2 , \, 4 x = 2 |
| Step 2: | x = - \dfrac{1}{2} , x = \dfrac{1}{2} |
x = - \dfrac{1}{2} only
x = \dfrac{1}{2} only
Neither
Both x = - \dfrac{1}{2} and x = \dfrac{1}{2}
Solve the following equations:
Solve the following equations:
Solve the following equations:
Consider the inequality \left|x\right| \leq 5.
First rewrite the inequality without absolute values.
Draw the solutions to the inequality on the number line.
Complete the following statement:
If c > 0, then \left|u\right| < c is equivalent to ⬚ < u < ⬚.
Draw the number line representation for each of the following:
\left|x\right| \leq 3
\left|x\right| \neq 2
\left|x\right| > 6
\left|x\right| \neq 6
\left|x\right| < 4
\left|x\right| = - 7
\left|x\right| \leq 9
\left|x\right| = 9
\left|x\right| \geq 5
\left|x\right| > - 5
Rewrite the following inequalities without using absolute values:
Write the following inequalities as absolute value inequalities:
Write an absolute value inequality that represents:
All real numbers x that are less than 8 units away from 0.
All real numbers x that are more than 8 units away from 0.
All real numbers x that are at least 2 units away from 8.
All real numbers x that are at most 5 units away from - 2.
Solve the following inequalities:
Solve the following inequalities:
Manufacturing Standards:
Cell phone cases have dimension requirements to ensure the phone will fit properly in the case. The manufacturing engineer has written the specification that the new length, n, of the case can differ from the previous length, p, by only 0.04 centimetres or less. The inequality is \left|n - p\right| \leq 0.04
Find the limits of the new length of a cell phone case if the previous length was 18.9 centimetres.
If a coin is tossed 100 times we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if h, the number of outcomes that result in heads, satisfies \left|\dfrac{h - 50}{5}\right| \geq 1.645 .
Solve the inequality.
Hence, complete the following sentence using integer values:
If a coin is tossed 100 times and the number of outcomes that result in heads is ⬚ or less or ⬚ or more, then the coin is unfair.
In a certain company, the measured thickness, m, of a helicopter blade must not differ from the standard, s, by more than 0.17 millimetres. The manufacturing engineer expresses this as the inequality \left|m - s\right| \leq 0.17.
Find the possible values of m if the standard, s, is 17.92 millimetres.