Algebra

Inequalities

Find values and regions that satisfy a condition.

Predict, test and explain

Step 1 of 3

Predict

Which values satisfy −2x + 1 ≤ 5?

2x+1≤52x + 1\leq5
−10-10
−8-8
−6-6
−4-4
−2-2
00
22
44
66
88
1010
boundary:  x=2\text{boundary}:\;x=2

Scroll the diagram sideways to see every label.

x≤2x\leq2

A filled endpoint is included; an open endpoint is excluded. Shading continues beyond the displayed number line.

InvestigateWhat happens when the coefficient of x becomes negative?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Solve this inequality.

−3x+2>11-3x+2>11

Hint

Subtract two first. Dividing by a negative number reverses the inequality sign.

Reveal answer

Divide by negative three after subtracting two; the boundary is excluded.

x<−3x<-3

Build an explanation

Hint 1

Find the boundary value as if the inequality sign were an equals sign.

Hint 2

Check which side satisfies the condition; strict signs exclude the boundary.

Hint 3

Reverse the inequality sign when dividing or multiplying by a negative number.

Worked example

This example uses fixed values, separate from the diagram controls.

Solve −2x + 1 ≤ 5.

  1. Subtract 1 from both sides: −2x ≤ 4.
  2. Divide by −2 and reverse the sign.
  3. The boundary is included because the sign allows equality.

Answer: x ≥ −2; a filled endpoint at −2 with shading to the right.

Connect this idea

Watch out: Dividing by a negative reverses the order; dividing by a positive does not.