Solve each inequality. Graph the solution set and write it in interval notation. 7(x-2)+x ≤-4(5-

step 1 All right, we need to begin to isolate for X. And the way we can start out doing that is by simp

Solve each inequality. Graph the solution set and write it...

Key Concepts

Graphing on a Number Line and Interval Notation

Graphing the solution of an inequality on a number line involves marking the region where the inequality holds true, typically using open or closed circles to denote exclusion or inclusion of endpoints. Interval notation is a concise way of representing these solutions, using brackets and parentheses to denote whether the endpoints are included or excluded from the solution set.

Inverse Operations

Inverse operations are methods used to isolate the variable in an inequality by performing operations that reverse the effect of addition, subtraction, multiplication, or division. When working with inequalities, it is important to remember that multiplying or dividing by a negative number reverses the inequality sign, which is a critical step in accurately solving the inequality.

Distributive Property

The distributive property is a key algebraic principle used to expand expressions where a term outside the parentheses multiplies each term inside the parentheses. This property is critical in simplifying expressions in inequalities or equations, ensuring that operations are correctly applied to each term in the expression.

Combining Like Terms

Combining like terms involves merging terms in algebraic expressions that have the same variable raised to the same power. This process simplifies the inequality by reducing it to a form that is easier to solve, making it possible to isolate the variable on one side of the inequality.

Linear Inequalities

Linear inequalities are expressions that involve a linear function and a relation (such as ?, <, >, or ?) instead of an equality sign. They allow us to describe a range of values that satisfy the inequality, rather than a single value as in equations. This concept is foundational in understanding how to determine solution sets that can be represented both on a number line and in interval notation.

Transcript

00:01 All right, we need to begin to isolate for x.

00:05 And the way we can start out doing that is by simplifying the left and right sides of this inequality statement.

00:14 So i'm going to do that by distributing this 7 that's in front of this set of parentheses.

00:21 So multiplying 7 times x will get me 7x minus 7 times 2, which is 14.

00:29 And we still have plus x because we haven't done anything with.

00:32 That yet is less than or equal to.

00:35 And we're going to do the same thing with this negative 4 by distributing to everything inside of the parentheses.

00:42 So negative 4 times 5 is negative 20.

00:46 And then this x is really a negative x.

00:48 So negative times a negative is a positive.

00:52 So essentially 4 times x minus 12.

00:59 All right.

01:00 So my next step could be combining like terms.

01:03 So on the left side i have 7 x's and 1x so together that would be 8 x minus 14 and that is less than or equal to on the right side we have a negative 20 and a negative 12 so we can think of that as like 20 plus 12 which will get us to 32 except our answer will just be negative plus 4x and now we can start getting all of our x is onto one side.

01:42 So i'm going to do that by getting rid of 4x on the right hand side.

01:47 And whatever you do to one side, you must do to the other.

01:51 So i will have 4xes left on the left side minus 14 equals negative 32, or not equals less than or equal to negative 32.

02:03 All right...

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