# Solving Linear Equations and Inequalities

## Vocabulary

### absolute value

for a number $x$, the distance from $x$ to zero on a number line, represented by $|x|$. If $x\geq 0$, then $\lvert x\rvert=x$. If $x \lt 0$, then $\lvert -x\rvert=x$.

### absolute value equation

equation that contains an absolute value expression

### absolute value inequality

inequality that contains an absolute value expression

property stating that the solution of an equation does not change after adding the same number to both sides of the equation

property stating that the solution of an inequality does not change after adding the same number to both sides of the inequality

### distributive property

property stating that multiplying an expression by a sum is the same as multiplying the expression by each term in the sum and then adding the products:
$a(b+c)=ab+ac$

### division property of equality

property stating that the solution of an equation does not change after dividing both sides of the equation by the same nonzero number

### division property of inequality

property stating that the solution of an inequality does not change if both sides of the inequality are divided by the same positive number or if both sides of the inequality are divided by the same negative number and the inequality sign is reversed

### linear equation in one variable

equation in which each term can be written as a number or a product of the variable and a number. The variable is not raised to a power, in the denominator of a fraction, or under a radical, such as a square root.

### linear inequality in one variable

inequality in which each term can be written as a number or a product of the variable and a number. The variable is not raised to a power, in the denominator of a fraction, or under a radical such as a square root.

### multiplication property of equality

property stating that the solution of an equation does not change after multiplying both sides of the equation by the same nonzero number

### multiplication property of inequality

property stating that the solution of an inequality does not change if both sides of the inequality are multiplied by the same positive number or if both sides of the inequality are multiplied by the same negative number and the inequality sign is reversed

### solution of an equation

value of the variable that makes the equation true, which means both sides are equal

### solution of an inequality

value of the variable that makes the inequality true, which means the comparison is valid

### subtraction property of equality

property stating that the solution of an equation does not change after subtracting the same number from both sides of the equation

### subtraction property of inequality

property stating that the solution of an inequality does not change after subtracting the same number from both sides of the inequality