Solving Linear Inequalities This presentation will show the steps for solving a linear inequality.

General Information To Get Started Interval notation is used to write inequalities. It is a shorthand for section(s) of the number line that are shaded in an answer. Example A: Reading left to right, this says the numbers in this inequality are those greater than – 4. On the number line, those numbers would start at – 4 on the left and go all the way to the end of the number line (which is represented by ∞). So from left to right the interval notation would look like this: A parentheses is used when something CANNOT be equaled. In the problem – 4 is not to be equaled and ∞ can NEVER be equaled!! Example B: Reading left to right, this says the numbers in this inequality are those less than or equal to 3. On the number line, those numbers would start at the far left end of the number line and move up to 3. Since - ∞ is at the far left end of the number line, and interval notation must go left to right, the answer would look like: A bracket is used when a NUMBER can be equaled. That is why 3 has a bracket. Notice - ∞ (just like ∞) can NEVER be equaled. That is why it has a parentheses.

Example 1: When both sides are divided by a negative number, the arrow Step 1: If the problem contains fractions or parentheses, clear them first. Watch your signs. Step 2: Combine like terms if possible. Step 3: Since the letter x appears on the right and left side of the inequality, add 3x to both sides. Then simplify. Step 4: Now, to solve for x, add 7 to both sides of the inequality. Then simplify. Step 5: To finish, both sides must be divided by -2. When both sides are divided by a negative number, the arrow MUST change directions!!!!! (NOTE: If both sides are divided by a positive value, the arrow does NOT change direction.) Step 6: Now, read your answer. Less than will look like a lazy “L”, greater than will not. So it says the answer are “the numbers greater than or equal to – 11/2”. Shading on a number line will help. Because there is an equal on the arrow, there will be a bracket on – 11/2: Notice that ∞ and - ∞ cannot be equaled so they receive a parentheses.

Example 2: Multiply the inequality on the left and right side by the common denominator that is 12. This will make all the fractions cancel. Now you have an inequality where you need to get x off the right side. You make that happen by subtracting 3x from both side of the inequality: Combine like terms on the left and right. This will give you a very simple inequality to solve by simply adding 24 to both sides. Notice you do not have to divide by any number here at the end of the inequality. So, read the answer as “numbers less than 29”. You may graph these numbers on a number line if you wish. Then write your answer in interval notation. Notice also that the less than arrow does not have an “equal” so parentheses (not brackets) are used to get the answer

Some inequalities involve solving for x in the middle!!! Example 3: To solve this inequality, get x by itself in the middle by first subtracting 2 from the left, middle and right. Simplify each part to get: Divide the entire inequality by positive 3. Notice, since 3 is positive, the arrows do NOT change direction!! Now that the inequality is solved, read the answer. Since x is “in between” – 2 and 3, the solution is all the numbers in between -2 and 3 on the number line. You may wish to shade those numbers on the number line to help with your interval notation. Notice the arrow next to -2 does not have an “equal”, and the arrow next to 3 does have an “equal”. So the interval notation answer is: (-2,3]

Definition: Remember that the absolute value of a number is Its distance from zero on a number line. Example 4: Absolute value is “distance from zero”. To find numbers whose distance from zero is less than or equal to 8, we would have to make sure those numbers fell between 8 and - 8. So, set up an “in between” problem like example 3. Write 3x – 1 must lie between – 8 and 8 this way: Now, solve the inequality with the same steps as Example 3: All absolute value less than problems will always set up the same way as an “in between” problem. Watch the interval notation and the answer is:

Example 5: Still remembering that absolute value is “distance from zero”, think about what is wanted in this problem. Find the numbers , represented by 2x + 7, whose distance is more than 11 steps from zero. Those would be numbers to the left of – 11 OR to the right of 11. Just remember “to the left of” means “less than” . Also, “to the right of” means “greater than”. Thus, we need to solve two separate inequalities to find all of the answers: Using steps from previous examples you would solve each of these: The answers are: “numbers less than – 9 “ or “numbers greater than 2”. Write those individually in interval notation, then join them using the union symbol “U” that represents the word “OR”: