Understand Solutions of Equations and Inequalities

Students learn that solving an equation or inequality means finding the value(s) of the variable that make the statement true. They use substitution to check whether a given value is a solution and understand that an inequality may have infinitely many solutions.

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Grade 6 · Equations and Inequalities · question 1 of 3

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20 rungs. A star at the top.

Find the dot whose number solves y − 2 = 2.

How it is taught

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A solution makes the equation true.

Substitute to verify a value.

Read it: Try each value

Which dots make x > 6 true? Put each number in for x: 3 > 6 is false, 6 > 6 is false, but 8 > 6 and 10 > 6 are true.

Read the number at each dot.
Put it in place of the letter and check if the sentence is true.
Keep the values that make it true: those are the solutions.

A solution is a value that makes the equation or inequality true.

An inequality can have many solutions: 8 and 10 both make x > 6 true.

The boundary needs care: 6 > 6 is false, because 6 is not more than 6.

To test a value, put it in place of the letter and check if the sentence is true.

Counting 6 as a solution of x > 6.

6 > 6 is false: x > 6 leaves 6 out, so 8 and 10 are the solutions here.

Which value makes 8x = 72 true: 6, 8, 9 or 11? — 9. Try each: 8 × 9 = 72, and none of the others makes 72.

x + 5 = 12 vs. x > 4

What does it mean to 'solve' an equation? It means finding the value of x that makes the statement TRUE.

Here's our equation: x plus five equals twelve. Let's hunt for the value of x.

Let's try x equals seven. We substitute seven in place of x.

Seven plus five is twelve. Twelve equals twelve — TRUE! So seven IS a solution.

Now let's try x equals eight. We substitute eight in place of x.

Eight plus five is thirteen. But thirteen is NOT twelve — so eight is not a solution.

Only x equals seven makes the equation true. An equation usually has just one solution.

Now look at an inequality: x is greater than four. Which numbers make THIS true?

Try x equals five. Five is greater than four — TRUE.

But ten works too. So does one hundred. Even four-point-one!

Grade 6 · Substitute and check what makes a statement TRUE

Every number to the right of four is a solution. There are infinitely many!

To check any solution: substitute and see if it's true. Equations usually have one. Inequalities can have infinitely many.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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