**High School Number Sense Lesson 49: Solving for X with Inequalities**

A couple of months ago we introduced algebraic equations with

**SolveX**and

**SolveXExpo**. Earlier this month we covered a special type with

**SolveXDenom**. Today we will find out what happens when we replace the

**=**sign with a

**>**,

**<**,

**≥**, or

**≤**sign. This concept appeared

**11 times**last year, with a median placement at

**question # 48**.

**Number Dojo Level: 224**

For a thorough introduction to (or review of) inequalities, I recommend

**Khan Academy**. For a

**"how do I work these problems on number sense tests?"**tutorial, keep reading.

Inequalities are basically solved like equations, with one caveat:

**flip the inequality sign when multiplying or dividing both sides by a negative number**.

**Example 1: If 5x - 7 < 13, then x < ___**

- Isolate x on the left by adding 7 to both sides.
**5x - 7 + 7 < 13 + 7**. - We now have
**5x < 20**. Divide both sides by 5 to get**x < 4**. - The answer is
**4**.

**Example 2: If 20 + 2x > 12, then x > ___**

- Isolate x on the left by subtracting 20 from both sides.
**20 + 2x - 20 > 12 - 20**. - We now have
**2x > -8**. Divide both sides by 2 to get**x > -4**. - The answer is
**-4**.

**Example 3: If 25 - 2x > 11, then x < ___**

- Notice that the problem has a > and a <. This is a clue to you that you'll need to flip the inequality sign.
- Isolate x on the left by subtracting 25 from both sides.
**25 - 2x - 25 > 11 - 25**. - We now have
**-2x > -14**. Divide both sides by**-2**and remember to flip the inequality sign around. - We now have
**x < 7**. The answer is**7**.

**Example 4: If 3x - 5 < 6x - 11, then x > ___**

- Stepping it up a notch...where x is on both sides. Let's isolate x on the right by subtracting 3x from both sides.
**3x - 5 - 3x < 6x - 11 - 3x**. - We now have
**-5 < 3x - 11**. Add 11 to both sides.**-5 + 11 < 3x - 11 + 11**. - We now have
**6 < 3x**. Divide both sides by 3. - We now have
**2 < x**. This is the same as**x > 2**. The answer is**2**.

**Example 5: The smallest integer x such that 4 + x > 10 is ___**

- Some of these problems require some math AND some logic. I know--torture!
- Isolate x on the left by subtracting 4 from both sides.
**4 + x - 4 > 10 - 4**. - We now have
**x > 6**. The answer can't be 6, so we have to add 1 to satisfy the logic. - The answer is
**7**. - (We can test our answer by plugging it back in to the inequality:
**4 + 7 > 10**.**11 > 10**, which checks out).

**Example 6: The largest integral value of x such that |2x + 5| ≤ 3 is ___**

- When we throw pipes (
**||**, also known as**absolute value**signs) into the mix, things get really interesting... We basically need to solve this one twice and then choose between the solutions. - Let's start with the positive.
**2x + 5 ≤ 3.**Isolate x by subtracting 5 from both sides.**2x + 5 - 5 ≤ 3 - 5**. - We are left with
**2x ≤ -2**. Divide both sides by 2. - We now have
**x ≤ -1**. This could be our answer. - Now let's get a little negative. Because of the absolute value (||), we need to set it up like:
**-(2x + 5) ≤ 3**. - Simplified, we have
**-2x - 5 ≤ 3**. Add 5 to both sides:**-2x - 5 + 5 ≤ 3 + 5**. - We now have
**-2x ≤ 8**. Divide both sides by**-2**. (Remember to flip the inequality for this one). - We now have
**x ≥ -4**. This could be our answer. - Combining the two solutions, we get
**x ≤ -1 and x ≥ -4**. In other words,**-4 ≤ x ≤ -1**. The question asked for the**largest integral value**, so the answer is**-1**.**THESE TAKE PRACTICE! So...**

**Here's a free worksheet to help you practice Inequality:**

inequality.pdf |

**Up Next for High School: EstHighPower**