Question

Four times a number decreased by 6 is less than$-2$. Define a variable, write an inequality, and solve for the number.

Short answer

Let x denote the number.

The inequality is $4x-6<-2$.

The number is less than 1 as the solution of the inequality$4x-6<-2$ is $x<1$.

Step-by-step solution

Step 1. Write the addition and division property of inequalities. 

The addition property of inequalities states that if the same number is added to each side of a true inequality, the resulting inequality is also true that is:

(i) If $a>b$, then $a+c>b+c$.

(ii) If $a<b$, then $a+c<b+c$.

The division property of inequalities states that if both sides of the inequality are divided by a positive number the sign of the inequality remains the same and if both sides of the inequality are divided by a negative number then the sign of the inequality changes that is:

(i) If$a>b$and cis a positive number then $\frac{a}{c}>\frac{b}{c}$.

(ii) If$a<b$and cis a positive number then $\frac{a}{c}<\frac{b}{c}$.

(ii) If$a>b$ and cis a negative number then $\frac{a}{c}<\frac{b}{c}$.

(iv) If$a<b$ and c is a negative number then $\frac{a}{c}>\frac{b}{c}$.

Step 2. Define a variable and write an inequality.

It is given that Four times a number decreased by 6 is less than$-2$.

Let x denote the number.

Therefore, the inequality is:

$4x-6<-2$

Step 3. Solve the inequality 4x−6<−2.

The solution of the given inequality$4x-6<-2$ is:

$$\begin{gathered} 4x-6<-2 \\ 4x-6+6<-2+6\left(\text{byusingadditionpropertyofinequality}\right) \\ 4x<4 \\ \frac{4x}{4}<\frac{4}{4}\left(\text{byusingdivisionpropertyofinequality}\right) \\ x<1 \end{gathered}$$

Therefore, the solution of the inequality$4x-6<-2$ is $x<1$.

Therefore, the number is less than 1.