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Expert-verified Found in: Page 323 ### Algebra 1

Book edition Student Edition
Author(s) Carter, Cuevas, Day, Holiday, Luchin
Pages 801 pages
ISBN 9780078884801 # Four times a number decreased by 6 is less than $-2$. Define a variable, write an inequality, and solve for the number.

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 $x<1$.

See the 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, then $a+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 c is a positive number then $\frac{a}{c}>\frac{b}{c}$.

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

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

(iv) If $a 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:

$4x-6<-2\phantom{\rule{0ex}{0ex}}4x-6+6<-2+6\text{}\left(\text{by using addition property of inequality}\right)\phantom{\rule{0ex}{0ex}}4x<4\phantom{\rule{0ex}{0ex}}\frac{4x}{4}<\frac{4}{4}\text{}\left(\text{by using division property of inequality}\right)\phantom{\rule{0ex}{0ex}}x<1$

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

Therefore, the number is less than 1. ### Want to see more solutions like these? 