Download Chapter 1 – Summary

Chapter 1 – Summary
Solve absolute value equations
Assumption: You can solve regular linear equations.
How do you solve, for x, an equation that looks like this?
3 * 2x  5  6  9
Strategy – Isolate the absolute value
Approach – Work PEMDAS backward
Step 1 – Move any subtraction or addition terms away from the absolute value. Follow
algebra rules to move these terms to the opposite side of the equal sign. Where there is
subtraction, then add to both sides of the equal sign. Or, where there is addition, then
subtract from both sides of the equal sign. [Note: never touch anything inside the absolute
value, until the entire absolute value is by itself.]
In the example, subtract 6 from both sides of the equation. The result would be:
3 * 2x  5  3
Step 2 – Move any division or multiplication terms away from the absolute value. Follow
algebra rules to move these terms to the opposite side of the equal sign. Where there is
division, then multiply on both sides of the equal sign. Or, where there is multiplication,
then divide on both sides of the equal sign. [Note: never touch anything inside the
absolute value, until the entire absolute value is by itself.]
In the example, divide both sides by 3. The result would be:
2x  5  1
Step 3 – If we follow PEMDAS, then we would look for exponentiation and parenthesis.
This will not occur in the typical Algebra 2 practice exercises.
Step 4 – Now that the absolute value is by itself, realize there are 2 possibilities. The
inside of the absolute value could be positive or negative. Demonstrate this by showing
the 2 equations that could exist.
In the example, the 2 equations would be:
2x  5  1
and
2x  5  1
Step 5 – Solve each one of the 2 equations generated in step 4. [Remember, we have
assumed that you know how to solve these normal linear equations.]
In the example, the solutions would be:
2x  5  1
and
2 x  5  1
2x  6
and
2x  4
x3
and
x2
We added 5 to both sides.
We divided both sides by 2.
Calculator tricks using a TI-83 or TI-84
Strategy – graph the absolute value equation and see where it crosses the x-axis.
Approach – convert the absolute value equation to a y= format.
Step 1 – Move everything to one side of the equal sign.
In the example, the result would be:
3 * 2x  5  6  9
3 * 2x  5  6  9  0
Subtract 9 from both sides
Step 2 – Replace the zero with y.
In the example, the result would be:
3 * 2x  5  3  y
Resolve  6 - 9 to - 3 and replace 0 with y
Step 3 – Enter this y= equation into the calculator and graph it.
In the example, the graph would look like:
Notice that the graph crosses the x-axis at x=2 and x=3. If you are not sure where the
graph crosses the x-axis, use the graph zero function of the calculator (step 4).
*** If the graph does not cross the x-axis, there is no valid solution to the equation. ***
Step 4 – Use the calculator zero function to find where the graph crosses the x-axis.
The commands are
2nd Calc
#2 (for zero)
Then position the spider (askeriques) near the zero you want to find. Use only the
left or right arrows.
[Note: the calculator only finds one zero at a time.]
When asked about the left bound, use the left arrow to move the spider.
Press enter
When asked about the right bound, use the right arrow to move the spider, past
the zero.
Press enter
When asked to make a guess,
Press enter
The answer is at the bottom of the screen. Specifically, the x value is the answer
for one of the zeros. The y value should be zero, or something very close to zero.
Repeat this process to find the second zero. This time remember to position the
spider near the second zero.
Solve inequalities
Assumption: You can solve regular linear equations.
How do you solve, for x, an equation that looks like this?
2x  5  9
Strategy – Isolate the variable.
Approach – Work PEMDAS backward
Step 1 – Use your skills in solving linear equations.
*** Remember – if you multiply or divide by a negative, the direction of the inequality
changes. ***
In the example, subtract 6 from both sides of the equation. The result would be:
2x  5  9
2 x  14
add 5 to both sides
x7
divide both sides by 2
Step 2 – Use set notation to write the answer calculated in step 1.
In the example, the result would be:
x7
or
[7,)
[ means to include 7 up to positive infinity)
Or
{x | x  7} Verbally t his is, ' x' such that x is greater th an or equal to 7
Calculator tricks using a TI-83 or TI-84
Strategy – graph the inequality.
Approach – graph and read the graph and/or the table.
*** The table will not give the exact answer, but will be close ***
Step 1 – Enter the inequality exactly into y=
In the example, the result would be:
y  2x  5  9
Step 2 – Graph y=
*** A straight line at y=1 appears above the valid x answers. ***
Look at the table. Valid answers have a 1 in the Y column.
*** The table only gives answers based on the “tblset” settings ***
*** Fractional answers or excluded points could be missed, be careful. ***
2nd example – compound inequality
3x  1  7 or 2x  9  7
Put this exactly into the calculator. Be sure to use the ‘Test’ and ‘Logic’ operators to
enter the inequalities and the ‘or’ logic statement.
The resulting graph will look like: