Thirty-Six

Unit 3 · Linear RelationshipsA3 · Inequalities · Structures

Inequalities & Absolute Value

Solve ranges and distance statements, including the sign flip and two-case traps.

  • 29 min
  • Unlocks: Pressure Tolerances
  • Desmos way

Ready when you are

7 steps · about 29 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

Finish the previous step to continue, or use “Show everything” above to explore.

Key terms

Boundary
An endpoint separating values that satisfy an inequality from those that do not.
Strict inequality
An inequality that excludes its boundary.
Absolute value
Distance from zero on the number line.

Step 2 of 7Try it firstDone

Try it first

Finish the previous step to continue, or use “Show everything” above to explore.

Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

Step 3 of 7LearnDone

The big idea

Finish the previous step to continue, or use “Show everything” above to explore.

See the structure

Solve a linear inequality like an equation, except multiplying or dividing by a negative reverses the inequality sign.

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

Absolute value is distance from zero. ∣x−h∣=r|x-h|=r gives two points h−rh-r and h+rh+r; ∣x−h∣<r|x-h|<r gives values between them.

Explore in Desmos

Locate an inequality boundary

The graphs meet at x=−4x=-4, the boundary of −2x+3>11-2x+3>11. Test x=−5x=-5 and x=−3x=-3 in the original inequality to determine that the solution lies to the left and excludes -4.

The calculator is live: change anything and see what happens.

Locate an inequality boundary

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

Loading Desmos…

Stretch to a 36

A strict inequality excludes its boundary; a non-strict inequality includes it. In Desmos, dashed and solid boundaries show that difference.

Trap

A believable answer to the wrong question

The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.

Step 4 of 7Worked examplesDone

Worked examples

Finish the previous step to continue, or use “Show everything” above to explore.

Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Reverse the inequality when dividing by a negative

Problem

Which inequality describes all solutions of −2x+3>11-2x+3>11?
Subtract 3: −2x>8-2x>8.
Divide by −2-2 and reverse the sign: x<−4x<-4.

Which case or inequality rule applies?

Which test value or distance check confirms the solution set?

Worked example

Example 2: Split an absolute-value equation into two cases

Problem

What values of xx solve ∣x−6∣=4|x-6|=4?
Distance 4 from 6 means x−6=4x-6=4 or x−6=−4x-6=-4.

Which case or inequality rule applies?

The solutions are x=10x=10 and x=2x=2.

Which test value or distance check confirms the solution set?

Faded example · you finish it

Example 3: Complete the setup and result

Problem

What is the greatest integer satisfying ∣x−5∣<3|x-5|<3?

Choose the setup that follows the stated relationship.

Choose the resulting value.

Desmos way

Graph an absolute-value solution interval

Type abs(x-5)&lt;3 to display the inputs less than 3 units from 5. The boundary values are 2 and 8, and strict inequality excludes both.
Graph x=7 and x=8. The line at 7 lies in the allowed band; 8 is on the excluded boundary, so 7 is the greatest integer solution.

Graph an absolute-value solution interval

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

Loading Desmos…

Step 5 of 7PracticeDone

Practice

Finish the previous step to continue, or use “Show everything” above to explore.

Step 6 of 7Spot the mistakeDone

Spot the mistake

Finish the previous step to continue, or use “Show everything” above to explore.

Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

Finish the previous step to continue, or use “Show everything” above to explore.
Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.